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A straight line graph 2020-02-26
From Penehafo:
I have a problem of drawing a straight line graph by the equation y=2x-1
Answered by Penny Nom.
A system of inequalities 2020-02-01
From Isaac:
Austin has x nickels and y dimes, having a maximum of 15 coins worth a minimum of $1 combined. No more than 4 of the coins are nickels and no less than 11 of the coins are dimes. Solve this system of inequalities graphically and determine one possible solution.
Answered by Penny Nom.
Investigating y = (-2)^x 2020-01-13
From Gonzalo:
This is not precisely a maths question, but it is formulated based on my maths curiosity. Fidgetting with my new graphic calculator, I started graphing things and had the idea to graph $y=(-2)^x.$ The result surprised me, and I thought a little bit about it, stored it on the back of my brain, and promised myself to look deeper into it someday.
Answered by Harley Weston.
y = |x| - 1 2018-11-30
From alexis:
what does this mean y=|x|-1
Answered by Penny Nom.
Graphing y = 2x - 1 2018-03-12
From Himanshu:
Hi I Himanshu I having 1 doubt on 1 question this question of graph how to draw y=2x-1
Answered by Penny Nom.
The graph of graph y= -2x-1 2017-11-11
From Natividad:
How do you graph y= -2x-1?
Answered by Penny Nom.
The distance between the origin and a moving point on a graph 2017-10-16
From Paulina:
Find the rate of change of the distance between the origin and a moving point on the graph of y=x^2 +1 if dx/dt=2 centimeters per second
Answered by Penny Nom.
A pie chart 2016-12-05
From vickie:
Determine the central angle needed to form a pie chart for the following housing characteristic data: 63% owner occupied, 27% renter occupied, and 10% vacant
Answered by Penny Nom.
A piecewise function 2015-04-25
From uday:

f(x)={x^2+3x+2 / x+2, x(not equal to)=-1
{4, x=-1
how to find domain and how the graph looks like

Answered by Penny Nom.
A circle graph 2015-04-06
From Davis:
i don't know how to answer this question. please help!

a survey about student government shows the following results

110 like the program
120 think the program is unnecessary
210 plan on running for student government next year

if this information is put in a circle graph,what is the central angle for the students who like the program?

Answered by Penny Nom.
A piecewise graph 2014-05-10
From Zoe:
Lightning Energy charges residential users for each unit of electrical energy bought from them each quarter, according to the scale below:
0-500 units cost 17 cents per unit
500-1500 units cost $85 plus 10 cents for each unit in the excess of 500
1500 or more units cost $185 plus 9 cents for each unit in excess of 1500
Sketch a graph showing the charge C as a function of U for up to 2000 electricity units

Answered by Penny Nom.
Graphing piecewise functions 2014-03-29
From Rayven:
Hi! I'm in eighth grade, taking ninth grade algebra 1. I'm confused as to how to graph piecewise functions. I know that you have already answered a question similar to this (I did my research first) but it didn't completely help me on my homework. I have to graph piecewise functions for the specified domains, and create a table for the absolute values. I know that two bars around a number means absolute value (two bars around -2 makes it +2) , but how do I graph and chart the absolute value for the following:

f(x)= |x+3 | for -5≤x≤3

And then graph and chart: (on a separate graph):

f(x)= {x if x≤0
{x+1 if x <0

thank you!

Answered by Penny Nom.
A circle graph 2014-03-06
From Caitlyn:
I need to find the measure of the central angle that represents the amount of time spent on each activity.
My question includes a circle graph with the following: Sleep 31%, Other 15%, Entertainment 18%, Errands 7%, Work 20%, and Food 9%.

Answered by Penny Nom.
The graph of y=2x+4 2014-02-18
From tyler:
y=2x+4 graph
Answered by Penny Nom.
The graph of f(x)=x^2-4 2014-02-07
From Allison:
Graph f(x)=x^2-4 and give the five point of the x/y intercepts.
Answered by Penny Nom.
What does y= f(x) actually mean? 2013-12-31
From John:

I don't understand how to pick coordinates for y=f(x).

I took a look at your answer to a previous question here:


What does y= f(x) actually mean?

Answered by Penny Nom.
How do I graph y=-2x-1? 2013-12-07
From cindy:
How do I graph y=-2x-1?
Answered by Penny Nom.
How can we draw the equation "x+y=1" in 3d? 2013-05-03
From shimaa:
how can we draw the equation "x+y=1" in 3d
Answered by Robert Dawson.
Solve sin 3x = -0.1254 with x between o and 360 degrees 2013-02-21
From David:
sin 3x = -0.1254 0
Answered by Harley Weston.
Solve sin 3x = -0.1254 with 0 2013-02-21
From David:
sin 3x = -0.1254 0
Answered by Harley Weston.
Plotting y=f(x)+2 2013-01-21
From Gavin:
how would you plot y=f(x)+2
Answered by Penny Nom.
A bipartite graph 2012-10-29
From A student:
Suppose that G be a bipartite graph with maximum degree of k.

Prove that:

1)Exists a K-regular bipartite graph that G be subgraph it(H)

Answered by Claude Tardif.
The number of solutions of sinx = x^2 + x +1 2012-09-01
From anamika:
number of solutions of

sinx = x^2 + x +1

Answered by Penny Nom.
Translation of a graph 2012-06-02
From Sean:
the information is thus: $f(x)=3(x+1)^2-2$ I wish to translate up 2 units and move 2 units right.
Answered by Penny Nom.
The graph of a quadratic function 2011-12-27
From Thomas:
sketch a quadratic function with zeros at -3 and 1
Answered by Penny Nom.
2 + f(x) and f(x) + 2 2011-11-21
From Beth:
Do you do y=2+f(x) the same way as if the 2 came after the equation such as y=f(x) +2 when graphing?
Answered by Penny Nom.
A step function 2011-08-22
From Giovanna:
An average school bus holds 45 people. Show a graph between the relationship of students who need bus transportation and number of buses required?
Answered by Robert Dawson and Penny Nom.
Linear inequalities 2011-03-25
From Alice:
OK. I'm in algebra 1 and the linear inequality is
I don't completely know how to graph y>2 and I'm getting really frustrated
Please help me in any way possible!
Thank You!

Answered by Penny Nom.
Points on a graph 2011-02-15
From Maggie:
Which set of ordered pairs can be used to graph y = -2x-1.

B. (-2,-5),(1,-1),(5,9)
D. (-5,9),(-2,3),(4,-9)

I cannot figure it out. Could you help me?

Answered by Penny Nom.
The equation of a line 2010-10-19
From Sava:
I have to write an equation and sketch a graph for the line that meets the given conditions:

A line with slope -15/5 that passes through the point (-2.5,4.5)

Answered by Penny Nom.
A taxi ride 2010-09-22
From Jack:
A taxi ride costs you $2 for the first km and 10 cents for each tenth of a km thereafter. Sketch a graph of the total cost $C(x) for a ride of xkm, for 0 < x < 2. Hint: The graph is not smooth.

How will I start this question off? I tried the table of values method, and then graphed it, but it comes out as smooth,

Answered by Stephen La Rocque.
The intersection of two graphs 2010-06-27
From Austin:
In depth explanation of how to find the intersection points of functions y=-4t+300 and y=-4.9(t-5)^2+300
Answered by Stephen La Rocque.
graph y=(2x^2-3x)e^ax 2010-05-31
From James:
graph y=(2x^2-3x)e^ax
Answered by Robert Dawson.
Which values of x satisfy (x-2) / (x+4) <7 2010-05-23
From Rocco:
Which values of x satisfy (x-2) / (x+4) <7
Answered by Harley Weston.
Modelling the roof of a house 2010-03-23
From Sandi:
This question has been haunting my dreams at night. It states the cross section of the roof of a house is modelled by the function y= -5\12|x-12|+5, where y>0 or y. Now I'm suppose to graph the function. b) Find the slope, height, length of sides, and base of the isosceles triangle. c)Explain what transformation must be applied to the graph of y=|x| to obtain the graph of y= -5\12|x-12|+5, where y>0 or y=0. 1st I put the equation into my calculator and graphed it. I got 10 for the height. We have had 2 sub teachers this past week. The first one told me the height should be 12 the other one told me 5. So I'm really confused. If I use my 10 units as height and the Pythagorean Theorem I get my sides to be 26 units the base 48 and my slope I calc to be 5/12. Both teachers are pretty sure I'm wrong and they are right but we all have different answers. Please help put this problem to rest so I can rest soundly at night. Thank you for your time.
Answered by Penny Nom.
Sketch the graphs of the following 2010-03-02
From musaf:
without detailed plotting of points,sketch the graphs of the following showing relevant information on the graphs:
a) y=(x-3)2 +5

Answered by Penny Nom.
The adjacency matrix of an undirected graph 2010-01-15
From Bhavya:
Let Cn be the undirected graph with vertex set V = {1,2,3,...,n} and edge set E = {(1,2), (2,3), (3,4),.... , (n-1,n), (n,1)}. Let An be the adjacency matrix of Cn.
a. Find the determinant of An.
b. Find (An)^2

Answered by Robert Dawson.
Using graphs 2010-01-04
From Armstrong:
Given that 50 litres of kerosine costs 6000 nira. Draw a graph and read off;
A: The costs of 15 and 34 liters.
B: The number of litres that can be bought for 2300 nira.

Answered by Robert Dawson.
f(x)=x+2sinx 2009-12-12
From amroziz:
for which values of x does the graph of f(x)=x+2sinx have horizontal tangent
Answered by Harley Weston.
The graph of f(x)=1/x -2 2009-11-29
From Kapilan:
Please answer the following question:
Sketch the graph of f(x)=1/x -2

Answered by Harley Weston.
The graph of f(x-2) 2009-11-19
From Kapilan:
Hi please answer the following question:
Given the function f(x)=x^2 sketch the graph of f(x-2)

Answered by Robert Dawson.
f(x)= (e^x) / [(e^x)+(ex^2)] 2009-11-10
From natalie:
I'm trying to graph the function, f(x)= (e^x) / (e^x)+(ex^2) [e to the x divided by e to the x plus e times x squared] I know that there aren't any vertical asymptotes, but is there a horizontal asymptote? and also, I'm stuck on finding the concavity for this graph. I tried to find f "(x), but it came out to be really long and I am not sure how to find the x values for f "(x) without using a graphic calculator. thanks, natalie
Answered by Chris Fisher and Harley Weston.
The intersection of two graphs 2009-10-30
From asma:
by using the suitable scales,draw both the graph of x/2 + y/4 =3 and y=2 on the same graph paper.Find the points of intersection of the two graph.
Answered by Penny Nom.
Graphing y=(4-x^2)^5 2009-10-25
From natalie:
I want to graph the curve of y=(4-x^2)^5 without using a graphing calculator. To do this, I'm suppose to find: domain, y and x intercepts, asymptotes, intervals of increase/decrease, local max/min, concavity and points of inflection. I got all the way to the step where I'm solving the concavity and I'm stuck. I found the f"(x) and it came out to be really large polynomial. I want to know how I can solve for the x of f"(x) without the use of a graphing calculator, when the polynomial has x^6 and x^8. Thank you so much, natalie
Answered by Harley Weston.
The intersection of two graphs 2009-09-02
From Jaqueline:
Find the points of intersection of the graphs of the equations

x^(2 )+ y^2=25 and 2x + y = 10

Answered by Penny Nom.
y=2x+1 and y=2x-1 2009-08-28
Need help of how to work this problems

y=2x+1 and y=2x-1

First of all what is the difference?

Second I think is easy just to work with the X y with the line in the middle

but how do you get the numbers for each side.

Answered by Penny Nom.
Graph 2x-3y+15=0 2009-08-20
From Diem:
Graph 2x-3y+15=0
Answered by Leeanne Boehm.
How many times does the graph of y=2xsquared -2x + 3 intersect the x-axis? 2009-05-27
From Henry:
how many times does the graph of y=2xsquared -2x + 3 intersect the x-axis?
Answered by Penny Nom.
Stairway construction 2009-05-02
From Olivia:
Hi, I'm Olivia and I'm in 8th grade studying algebra 1. I'm having some trouble with my math project. It has to do with rules of stairway construction. It says that there are two generally accepted rules to building stairs. Rule 1: x+y=43 and x+y=46 (cm) Rule 2: 2y+x=61 and 2y+x=64 (y is the riser and x is the tread length) I cant figure out how to graph them...it says they're supposed to intersect to make a square which is like the safety zone for building stairs (the stairs' rise and tread lengths are plotted as points) any points outside of that square are considered unsafe. Did i explain it ok? Thanks for your help. --olivia
Answered by Penny Nom.
The graph of y = 4x^-4 2009-04-07
From lyric:
tell whether the graph opens up or down.write an equation of the axis of symmetry

Answered by Penny Nom.
A simple digraph 2009-03-23
From Shabkhal:
Show that the sum of in degrees of all the nodes of a simple digraph is equal to the sum of out degrees of all its nodes and this sum is equal to the number of edges of the graph.
Answered by Victoria West.
An equation for a line with a changing slope 2009-03-21
From Ben:
Is there anyway to write an equation for a line with a changing slope. e.g. A cell [hone company has a initial fee of 50$ paying for the first 100 minutes, the rate then increases to 10 cents per minute, then at 500 minutes the rate is 5 cents per minute.

If so please tell how.

Answered by Penny Nom.
The graph of y = -6 2009-02-21
From Barb:
I was given the problem that says "graph the line y = -6" how do I find all the set points that have a y coordinate of -6? I am so lost lost and have no clue how to start. Can you help me please? thanks.
Answered by Penny Nom.
The intersection of two lines 2009-01-30
From kaylee:
how do you find and graph the intersect of these two equations: y=2x-4 (and) 2y=x thanks, kaylee
Answered by Stephen La Rocque.
y = |x| - 4 2009-01-28
From tiana:
how do you find where to plot Y in a problem that states y=|x| -4?
Answered by Robert Dawson and Penny Nom.
Graphing a parabola 2009-01-27
From Kimberly:
I need help with this parabola: graph y=5x^2-5x-6. I am not really understanding how to graph parabolas in general. Can you help me?
Answered by Penny Nom.
Graphing 2008-12-09
From Natasha:
Answered by Robert Dawson.
Graphing Rational Functions 2008-12-05
From Bahra:
The question in my homework is: Find all vertical, horizontal, and slant asymptotes, x- and y-intercepts, and symetries, and then graph each function. Check your work with a graphing calculator. y=3/x+3 meanng: ( y= 3 over x+3)
Answered by Janice Cotcher.
Scaling a logo 2008-10-20
From Carl:
Hello! I hope that you can help me out here. I've been trying to figure this out for awhile now and can't come up with the answer logically. I am a graphic designer by trade and need to figure out how large my client's logo should be. The logo will need to be sized proportionally with the surface area of the page. The logo will always be sized at 1.2938(w) x .75(h) on a letter size page (8.5 x 11).
My question is...
When you increase the size of the page (not always proportionally sized - dimensionally) how do figure out the NEW size of the logo. I could have a page 20" x 40" or 30' x 4'... or ANY size really. I believe I have the new surface area of the logo but can't find the new dimensions.

Answered by Harley Weston.
Concavity and the second derivative 2008-10-15
From Christina:
I'm having trouble solving for a second derivative for the following graphing question.

f(x) = (X^2+2x+4)/2x

using the quotient rule, I found:
f'(x) = (x^2-4)/(2x^2)

however, using the quotient rule again I can't seem to solve it (concavity):
f''(x)=[(4x^3-(4x^3 -16x)]/4x^4

and making the equation equal to zero result in 0=4 which doesn't seem to make sense...

Answered by Penny Nom.
Graphing a piecewise function 2008-08-12
From Shanti:
how do you graph a step function or piecewise function such as
u^2 if u is greater than or equal to -2 and less than 1
2- u if u is greater than or equal to 1 and less than 4

Answered by Penny Nom.
Two equations in two unknowns 2008-08-01
From James:
Question from james, a student:


My math teacher says that it is (2,-4)

Can someone please explain how to figure this out by graphing?

Answered by Victoria West.
Graphing Using Double Angle Identities 2008-07-16
From Hodan:
the Question is: Describe how you could use your knowledge of Double angle formulas to sketch the graph of each function. Include a sketch with your description.A) F(x)=sin x cos x B)F(x)=2 cos(squared)x C) F(x)= tan(x) (divided) by 1-tan(squared) x
Answered by Janice Cotcher.
Domain & Range of a Periodic Function 2008-07-11
From Michelle:
The depth, w metres, of water in a lake can be modelled by the function, w=5sin(31.5n+63) +12 where n is the number of months since January 1, 1995. Identify and explain the restrictions on the domain and range of this function.
Answered by Janice Cotcher.
A point on a graph 2008-07-08
From Rita:
If (2, b) is a point on the graph of y = (x^2) + 4x, find the value of b.
Answered by Leeanne Boehm.
Points on a graph 2008-07-03
From Rita:
Dertermine if the given 3 points are on the graph of the equation.

Equation: y = (x^3) - [2(sqrt{x})]

Points: (0, 0); (1, 1); (1, -1)

Answered by Penny Nom.
A piecewise equation 2008-06-16
From John:
AT&T charges $39.99 for 450 minutes per month. It charges 45 cents per minute over the 450 minutes. How can I make a piecewise equation of this problem?
Answered by Penny Nom.
The wedges in a circle graph 2008-05-18
From Libby:
I don't understand how to do this word problem: The cafeteria workers distributed a survey to the student body asking students to pick 1 from a list of 5 choices for their favorite lunch. The circle graph below gives the results of the survey for the students who responded. What is the measure, in degrees, of the central angle for each choice?

And there is a pie chart that has food choices with these percentages: 25%, 20%, 15%, 10%, and 30%

Answered by Leeanne Boehm and Penny Nom.
Identify the graph 2008-05-05
From A parent:
Refer to the graph given below and identify the graph that represents the corresponding function. Justify your answer.
y = 2x
y = log2x
I can figure out what line on the graph is which equation but unsure what is meant by justify your answer.

Answered by Penny Nom.
How would I graph: x - 2 divided by x^2 - 4? 2008-03-19
From Sean:
How would I graph: x - 2 divided by x^2 - 4 using a method in extreme values of functions or completing the square

x - 2/ x^2-4

Answered by Stephen La Rocque.
Graph the equation P(x)= x^3 + x^2 -12x 2008-03-18
From Sean:
I want to graph the equation : P(x)= x^3 + x^2 -12x I was thinking that I would combine the x-values, but then I could not use the complete the square technique. Would I have to just solve the second half of the equation and then graph x^3 seperately
Answered by Penny Nom.
Pythagorean equation 2008-02-27
From Judy:
X^2 + Y^2 = 5^2
Graph all ordered pairs in the coordiante plane that satisfy the Pythagorean equation, x squared plus y squared equals five squared

Answered by Stephen La Rocque.
How do I graph y = -2x^2? 2008-02-12
From Max:
How do I graph y=-2x^2?
Answered by Penny Nom.
Graphing a parabola 2008-01-16
From Sean:
How do I find the roots and describe the roots when graphing a parabola?

y = - x^2 - 4x -3

Answered by Stephen La Rocque.
Pie graphs 2008-01-15
From amarilis:
explain why circle graphs are called pie graph or pie charts
Answered by Stephen La Rocque.
Finding the equation of a line from its graph 2007-12-25
From ken:
I have a graph and it is asking me to find the slope of each line, and then write its equation. the graph show a negative slope. Would appreciate the answer if you could show me how to do this question.
Kind regards: Ken

Answered by Stephen La Rocque.
Pictographs 2007-12-11
From eniyah:
The title of my pictograph is Favorite pets. I have two columns. in my first column is the kind of pet and the second column is the numbers of students, the dogs have 16 student that like them and cats have ten students the likes them. each pictures equals 2 stodents my question is. i have to draw paictures to show the number of students for each row. the key says each symbol equals 2 students. since 16 divided by 2 is 8 can you show me how to make this in a graph
Answered by Stephen La Rocque.
Plotting a function 2007-12-06
From Paula:
My son has a problem that says plot the point: 2x-3y=9
I have explained to him that there must be a function for y in order to plot that point, but he doesn't think so. The question above that one says find the function for y, and I assume we should use that function. Am I wrong? Can we plot a point without a y function.

Answered by Stephen La Rocque.
Two lines 2007-11-18
From Kate:
represent qraphically the equations of these lines and their intersecting points

x-y=2 3x+y=11

Answered by Penny Nom.
Area of a triangle formed by three points on a graph 2007-10-26
From Betty:
My question is 'find the are of a triangle whose vertices have coordinates (3,5),(6,-5), and (-4,10)
Answered by Stephen La Rocque.
A graph with 100 vertices 2007-10-24
From amarjeet:
let g be a graph with 100 vertices numbered 1 to 100. Two vertices i and j are adjacent only if i-j=8 or i-j=12. The number of connected components in g are:
1. 8
2. 12
3. 4
4. 25

Answered by Penny Nom and Victoria West.
x = y^2 and x = -4 y^2 2007-09-11
From Jil:
My question is when dealing with parabolas, x=y^2, so that they are sifted on their side you could say, what will happen to the graph if you change it to x=-4y^2. I understand that the - flips the graph in the other direction but can you simply just plug in numbers and increase the stretch of the y?
Answered by Penny Nom.
What feature on a graph shows the solution? 2007-08-19
From San:
To solve the equation 2sinx+square root 3=0, what feature of the graph, in the form y=asinx+b would show the solution? Draw a graph to show the solution. thanks.
Answered by Stephen La Rocque.
Explain how you can shift a parabola up two units 2007-08-14
From Marsia:
Explain how you can shift a parabola up two units (y axis).
Answered by Stephen La Rocque.
Paying off a loan 2007-08-08
From Iona:
I borrowed $6000. 00 from a friend to pay school fee. He said I should not pay back any interest but pay $350.00 monthly after graduation. (1) Make an equation from the above (2) make a graph (3) what is the relationship between the months after graduation and the loan? (4) what is the horizontal and vertical intercept relate to the problem?
Answered by Stephen La Rocque.
The graph of a log function 2007-07-30
From Marsia:
How would you describe to another student what the graph of a logarithmic function looks like?
Answered by Penny Nom.
Period of a sum of trig functions 2007-06-17
From Aakash:
the period of the function f(x)=cos3x+sin4x+tan4x
Answered by Stephen La Rocque.
Parabolas 2007-05-16
From Andy:
How do you write an equation of a parabola that opens to the left with a vertex of (2,-6)?
Answered by Stephen La Rocque.
Finding a parabola's equation by looking at its graph 2007-04-30
From Kenzie:
The graph shows an arrow going upward crossing at the -2 on the x line and crossing the 3 on the x line and the vertex on the -6 on the y line.
Answered by Stephen La Rocque.
A sketch of a quadratic equation 2007-04-14
From Fee:
A ball is thrown upward from the roof of a 25m building. The ball reaches a height of 45m above the ground after 2s and hits the ground 5s after being thrown. Use the fact that the realtion between the height of the projectile and time is quadratic to draw an accurate graph of the relation on graph paper.
Answered by Penny Nom.
Graphing a line 2007-03-23
From mitchell:
Graph each equation. y = 2x + 4
Answered by Jaymi Peterson and Haley Ess.
A pictograph 2007-03-14
From Angie:
I am confused! I have to include a title, labels, and a key: Here is the problem

Snickers 45
Kit Kats 65
Nerds 30
Kisses 50
Suckers 75

With the above information please help me make a candy pictograph

I would gladly like your help as my child is a second grader and can not tell me anything about a pictograph

Answered by Penny Nom.
Statistics data 2007-03-08
From Parnini:
The following table shows the expenditure on different household items by a family. Draw a line graph for the following data:
Items : Food Rent Cloth Education Saving Others
Expenditure : 2000 2.500 1.500 1.200 4000 3.500

Answered by Stephen La Rocque.
y = x^2 - 2 2007-01-08
From Mike:
Graph the quadratic equation: y=Xsq-2
Answered by Penny Nom.
A linear system 2007-01-04
From Harold:
System A (4, 2/3)
2x-3y=6 and x-3y=2

I approached this problem using the addition method.

Could i have found the answer using the graphing method?

Answered by Penny Nom.
y = |x| - 10 2006-11-03
From Pam:
what is the process for finding the vertex point for an equation such as this, y= |x| - 10.
Answered by Penny Nom.
Plotting a linear equation 2006-10-09
From Jennifer:

I am having problems explaining how to do these problems to my child .


graph the equation: 2y+x=-3

Answered by Penny Nom.
Find the points of intersection 2006-08-18
From Ingrid:

The question says find the points of intersection of the graphs of the equation, and check your results analytically.

x squared + y squared=25
2x+ y=10

Answered by Stephen La Rocque.
A hamiltonian circuit 2006-07-24
From Tom:

1)Prove that every simple not directed graph with 21 vertices and 208 edges has
a hamiltonian circuit but not an Euler circuit.

(I have proved that if we give the 2 edges for each vertices then after a few
steps I have 21 vertices of odd degree and 19 edges.So in the end I have 2
vertices of odd degree.So I have not an Euler circuit.But how can I prove that
there is a Hamiltonian one?)

2)If G is a tree and has a virtex k degree,prove that it also has at least k
vertices degree 1.

Answered by Penny Nom.
What will the graph of y=x2 - 2x - 3 be? 2006-06-17
From Byrony:
What will the graph of y=x2 - 2x - 3 be?
Answered by Steve La Rocque.
How do you graph 3x + 6y = -9 by axis intercepts? 2006-05-22
From Crystal:
How do you graph 3x + 6y = -9 by axis intercepts?
Answered by Penny Nom.
Geometric sequence and basic functions (graphs) 2006-04-20
From Marlene:
Which of the basic functions is related to the geometric sequence: Linear, Quadratic, Rational, or Exponential? Can you give me an example of how it would be used in normal life?
Answered by Stephen La Rocque.
An epicycloid 2006-04-10
From Sharon:
What is the name of the curve formed by a point on the circumference of a circle that rolls on the outside of a fixed circle? This curve is used in the study of gears.
Answered by Stephen La Rocque and Penny Nom.
A table of values 2006-03-28
From Cindy:
Create a table of values for each of the following functions: y=5x, y=52, and y=5x. I dont know to do this.
Answered by Stephen La Rocque.
The graph of y = 2x + 1 2006-02-23
From Geneva:
I'm having trouble with graphing equations like y=2x+1 which is my homework I'm not sure how to graph it. I'm in the 8th grade taking a 9th grade course algebra I.
Answered by Penny Nom.
Whose photo is he looking at? 2006-02-19
From Adeyeri:
Looking at a picture a man says, that the father of the person (in photo) is the son of my father. whose photo is he looking at?
Answered by Penny Nom.
Broken-line graphs and histograms 2006-02-16
From George:

1. What is the main difference between a broken-line graph and a histogram? Both represent continuous variables.

2. What is the correct way to read a multiplication array: x-axis first and then y-axis, other way around or it doesn't matter?

Answered by Penny Nom.
Intercepts and slope 2006-01-15
From Anthony:
Given -1.7x + 0.8y=7.1:

(a) state the intercepts and slope for the graph

(b) Draw the graph

Answered by Penny Nom.
Graphing two lines 2005-11-20
From A student:
About y = 5x + 24 and y = -3x - 8
1. Graph both equations on the same set of axes. When you made your graph, how did you choose the range of x and y values?

2.Find the point of intersection for the graphs.

3. Test the point of intersection you found by substituting its coordinates into the equations. Do the coordinates fit the equations exactly?

Answered by Penny Nom.
Open dots and closed dots 2005-09-29
From Cynthia:
When graphing the solutions of an inequality, what is the difference between an open dot and a closed dot?
Answered by Penny.
Graphing a linear inequality 2005-08-26
From Gina:
When graphing a linear inequality, how do you know if the inequality represents the area above the line?
Answered by Penny Nom.
Graphing an inequality 2005-05-02
From Janice:
I need to know how to graph inequalities. I need to know just abot everything. From graphing an equation to shading.
Answered by Leeanne Boehm.
Graphing inequalities 2005-02-15
From Melanie:
Solve and Graph the following variable inequalities:





Answered by Penny.
A graph with certain properties 2004-11-22
From A student:
i was asked as a question in coursework to sketch the graph with the following characteristics:
a double root at -3
a pair of imaginary roots
an x-intercept at 6
a root at 4 which is not a double root

Answered by Penny Nom.
The points of intersection of two graphs 2004-11-05
From Benjamin:
How do I find the points of intersection of the two functions:
1) y =
2 - e-x 2) y = 1 + x2

Answered by Harley Weston.
f(x) = x-4 if x<2 2004-09-21
From Dani:
Hi, my name is Dani. I'm in tenth grade Algebra 2, and I'm learning about a lot of different kinds of functions, and I can't quite seem to understand how every part of the piecewise function works. I understand most of it, I think, but I've been looking at an example in my math book, and I'm not sure how they got the answer they did. The part I don't understand reads: "Graph f(x) = x-4 if x<2" and "Identify the domain and range." The picture in the book shows an open circle at (2,-2), with an arrow going down and to the left with the slope being one. How did they get the open circle to be at (2,-2), and why does the arrow point where it does?
Answered by Penny Nom.
The intersection of two graphs 2004-07-28
From JJ:
Is there a way to find the intersections of these graphs algebraically?

x^2 + y = 4 & 2x - y = 1

I got (1.45, 1.9) and (-3.45, -7.9) with a graphing calculator.


y = 3.29x & y = 5.5(x^0.5)+ 10000

I got x at 3133 with a graphing calculator.

Answered by Penny Nom.
The commuter 2004-07-14
From JJ:
A student who commutes 27 mi. to college remembers, after driving a few minutes, he forgot a term paper. Driving faster than usual, he returns home, gets the paper, and once again starts to school. Sketch the graph of the student's distance from home as a function of time.

Does it mean that the student drives at normal speed, drives faster to pick up the paper, and resumes normal speed to head back, or that the student drives at normal speed, then drives faster the rest of the way to college?

Answered by Penny Nom.
The intersection of two graphs 2004-03-31
From Benjanim:
How do I find the points of intersection of the two functions:

1) y = 2 - (e^-x)
2) y = 1 + (x^2)

I know that I have to set them equal to each other, but I can't solve for x. Please help :-)

Answered by Penny Nom.
The degree sequence of a graph 2004-03-31
From Ali:
How can i find out {6,6,5,5,5,3,2}is not the sequence of the points of a simple graph,without drawing it?
Answered by Penny Nom.
Graphing lines 2004-02-08
From A student:
how do you solve y=1\3x+3 and y=1\3x-3 by graphing?
Answered by Penny Nom.
Intercepts 2003-11-04
From A student:
I am totally confused on how to determine intercepts of an equation and then to graph the line. For example, how to determine the intercepts of 2x-3y-12=0.
Answered by Penny Nom.
Trees 2003-10-28
From Ali:
Is there any formula for counting the number of trees (graphs)?
Answered by Denis Hanson.
A parabola 2003-10-24
From Delores:
Given the vertex (4, -2) y intercept = -6 find if/where the parabola crosses the x axis?
Answered by Penny Nom.
The sketch of a graph 2003-10-07
From A student:
I was wondering how do you figure out if a graph has a horizontal tangent line. One of my homework problem was to sketch the graph of the following function; (4/3)x3-2x2+x. I set f''(x) ( the second derivative) of the function equal to zero and got the inflection point:(1/2,1/6). Also i am having trouble finding the concavity for x>1/2 and x<1/2, i am getting a different answer from the back of the book, the graph i draw looks completely different from the correct answer.
Answered by Penny Nom.
Functions, graphs and derivatives 2003-10-05
From Jathiyah:
I wanted to know how would you tell (on a graph diplaying two funtions), which funtion is the derivative of the other?
Answered by Walter Whiteley.
Graphing a piecewise function 2003-08-24
From Amber:
How do i begin to graph a piecewise function, absolute function or step function?
Answered by Penny Nom.
Two intersecting graphs 2003-04-23
From Patty:

a) graph the equation

x2 - y - 4 = 0

x2 + y2 = 9
on the same set of coordinate axes.

I did not have a problem with this.

The problem is part (b) of the question ask: Find all solutions of the system in part (a) algebraically. Express answers in decimal form, accurate to two decimal places.

Answered by Penny Nom.
y < 2x + 1 2003-03-01
From Erika:
how can I solve and graph y<2x+1 and which side in the graph should i shade after i finish solving my inequality?
Answered by Penny Nom.
y = 1 - sin(x + 60) 2002-12-10
From Eman:
Sketch the graph of y = 1 - sin(x+60). for 0 <= x<= 360, giving the coordinates of the maximum and minimum points and the pints where the curves crosses the y axis.
Answered by Penny Nom.
Performance reviews 2002-11-21
From Lara:
Perhaps you can help.

He's doing performance reviews at work.

The average performance review is 3.92 on a 5.0 scale. This person would receive a 3.5% raise for next year.

The highest performance review is 4.9 and this person would recieve a 6.0% raise for next year.

How do I solve for the other performance review numbers.

Answered by Penny Nom.
Intercepts 2002-09-04
From Tamara:
Why is it that the x intercept the value of y is 0 and at the y intercept the value of x is 0?
Answered by Leeanne Boehm.
A schedule for a fantasy football league 2002-09-01
From Steve:
i am in a ten(10) team league, split evenly into two five team divisions. we play a thirteen(13) game schedule and would like to play each team in your own division twice, and each team in the other division once. according to my rudimentary mathematics that equals 13 games. a good schedule should look somthing like this: team 1-5 = division 1 team 6-10 = division 2 team 1 should play teams 2-5 twice and teams 6-10 once.
Answered by Claude Tardif.
When does an inequality represent the area above the line? 2002-08-25
From Tamara:
How do you determine that an inequality represents the area above the line and when do you include the line in the solution?
Answered by Penny Nom.
A polynomial 2002-04-14
From A student:
A polynomial function is described by the following conditions: f(x) has two real zeros at x= -2 and x= 1, each of multiplicity 2 f(x) has two complex zeros as x goes to -infinity, f(x) goes to +infinity as x goes to +infinity, f(x) goes to +infinity f(x) has a y-intercept at (0,2)

Sketch a polynomial function that satisfies the above conditions.

Write an equation for function f(x)

Write another function g(x) that also satisfies the above parameters.

Answered by Penny Nom.
x and y-intercepts 2002-03-10
From A student:
I have the problem- f(x)= X-5/X2+X-6 and I have to find the vertical asymtope, horizontal asymtope, x-intercept, and y-intercept. And graph the problem. I am having problems finding the y-intercept.
Answered by Harley Weston.
Applied geometry 2001-11-02
From Jenny:
Where can I find some hands-on activities for my Applied Geometry classes? I want to do more activities with them that allow us to get out of the classroom. However, I want to use activities that use only inexpensive equipment because I usually buy the equipment myself.
Answered by Walter Whiteley.
Double bar graphs 2001-09-09
From Leslie:
My daughter has a homework problem in which she is graphing data from two frequency tables onto a double bar graph. The graph shows how many students worked in a program in two different years, grouped by age. In the first age group, one of the years has no students working, while there were students in that age group in the other year.
Answered by Penny Nom.
Graphing equations 2001-05-01
From A student:
If, for example, 4x2-12x+9=0 and -4x2+12x-9=0, which I'm assuming it does since you can derive that equation from the first, why do those two equations have different graphs?
Answered by Penny Nom.
f(x), f(x) + 2, f(x +2) 2001-04-04
From Monica:
Could you explain to me how one should go about graphing functions such as f(x), f(x+2), and so on. Also, how should you explain things such as constants and relationships among functions?
Answered by Penny Nom.
Graphing F(x) = 3x 2000-11-06
From Jose:
graph the exponential problem F(x)=3x
Answered by Harley Weston.
Graphing an inequality 2000-08-02
From Lori:
How do I find x and y and graph this problem

3x + y < 5

Answered by Penny Nom.
Graphing a linear function 2000-05-17
From Chelsea:
I need help with grahing linear functions.If you could e-mail me back the basics and how tos I would be much appriciative.
Answered by Penny Nom.
A matrix equation 2000-05-14
From A student:
Right now, we are dealing with matrices and we are supposed to solve the following problem on our graphing-calculators: 2a+3b-4c+d=20

Answered by Penny Nom.
Graphing cube roots 2000-04-27
From Heather Jones:
What do the graphs of cube roots look like?
Answered by Walter Whiteley.
Functions 2000-01-23
From Tara:
Hi my name is Tara, I have two math problems that I need help with in my calculus math class.

  1. If f(x)= x - 2 show that (x+3)f(x)-(x+2)f(x+1)+4=0

  2. Graph this function and use the graph to determine the range y=2x2 - 8x - 3

Answered by Harley Weston.
Bar graphs and histograms 2000-01-11
From Raeluck:
What is the difference between a bar graph and a histogram?
Answered by Harley Weston.
Factoring by graphing 2000-01-11
From Dylan Kirkwood:
How can you factor a quadratic equation by graphing it?
Answered by Penny Nom.
Y-intercept 2000-01-03
From Kayla Boylan:
How do you use and find the y-intercept?
Answered by Penny Nom.
Graph of a sine function 1999-12-23
From Pierre:
period: 540
Phase shift: 60 degree,right

I am ask to right the equation: sin 2/3 (value -60degree)

When I am asked to graph the equation, the period is mixing me up.
Answered by Harley Weston.

Crossing number 1999-11-06
From Christian:
The crossing number of a graph G, denoted cr(G) is defined to be the minimum number of (pairwise) crossings of edges among all drawings of the graph in the plane. For example, cr(K5)=1 and cr(K3,3)=1.
What is cr(K7,7)?

I figured out that the answer is 81.

Now I am trying to figure out if K7,7 can be drawn in the plane with less than 81 crossings?

I'm not sure how to approach this one. Other than actually drawing it out and checking by trial and error, I am not sure how to approach this problem. Please help!
Answered by Denis Hanson.

A pictograph 1999-10-28
From Ali:
Please tell me how to make a pictograph.I am 2 grade student.
Answered by Penny Nom.
Broken line graphs 1999-10-19
From Lori Samara, Kristi Cameron and the 4th grade students:
Students in Ms. Cameron's grade 4 class were wondering if a broken line can be used anywhere in the range on the y - axis when creating a broken line graph? They understand that the broken line can be used at the 0 on the broken line but what if there is a large gap in the data from 100 - 200? What do we do? The text book and mathematics dictionary do not answer these questions.
Answered by Penny Nom.
Intercepts 1999-09-24
From Cassandra:
My book says to find where the X and Y intercept, i dont understand who to do this problem. Can you please help here is the problem. it didnt quite explain the instructions.
Answered by Penny Nom.
Stem and Leaf Plot 1999-09-14
From Jeanette Sovick:
My 5th grade son brought home a math paper, the title of which reads, Reading Stem-and-Leaf Plots...can you explain this so I can explain it to him...There is no book, his teacher just sent this practice sheet home for him to complete and I have no clue!
Answered by Penny Nom.
Graphing a linear equation 1999-05-27
From Dylan Bradley:
Im in Grade ten and am doing linear equations, I cann't figure out how to make a chart for questions like y = -2x + 3
Answered by Penny Nom.
Dotted graph paper 1999-04-08
From Bridget Winward:
A teacher at our school is trying to locate dotted graph paper online or in print. His class would like to make three dimensional, geometerical drawings.
Please let us know if you have a good source.

Answered by Jack LeSage.
Graphing the Derivative 1999-01-18
From Milena Ghebre:
This question has been nagging me for sometime now.

Is there a way of finding out the derivative of a function, just by looking at the graph of it?
Answered by Walter Whiteley.

Complex numbers and the quadratic formula 1998-12-25
From Richard Peter:
My age is 16, and my name is Richard. My question relates to the topic complex numbers & the quadratic formula.

I would like to know how to solve quadratic equations in which the discriminant is less than 0 (i.e. we get two complex solutions to the quadratic)

3x2+2x+5 = 0

and how mathematicians like euler contributed to this field. If it would be possible I would also like to know how this type of quadratic equations can be graphed?
Answered by Harley Weston.

Intersection of planes 1998-11-22
From Dave Rasmussen:
I am a teacher of secondary mathematics with a question about the uses of Three Dimensional Co-ordinate Geometry. I have been teaching my students to write equations of planes and lines, - to find the intersection of these and the distance between them. What I am having difficulty finding are good applications of these techniques to "real world" situations. Can anybody help me?
Answered by Walter Whiteley and Harley Weston.
Points on a Graph 1998-10-03
From Nouver Cheung:
If the point P(-3,2) is on the graph of y=f(2x-1), what point must be on y=f(x)-3?
Answered by Harley Weston.
Graph question 1998-05-12
From Rose Seminary:
Why is the point of intersection of two lines the solution to the corresponding system of equations?
Answered by Penny Nom.
Graph Distortion 1998-02-19
From Dana Steffan:
What exactly is graph distortion? I have to explain it for a project and I can't seem to find anything on it.
Answered by Penny Nom.
The two-digit numbers from 10-99 1998-01-21
From Alexis Riddle:
My name is Alexis. I'm a student in 8th grade and I'm taking Algebra I Honors. My question is: In a string of numbers, two adjacent digits are considered as a two-digit number. For instance, the string 11012 contains the numbers 10, 11, and 12. What is the number of digits in the smallest string that contains all of the two-digit numbers from 10-99? Please help and thank you for any assistance you can give me.
Answered by Penny Nom.
Quadratic Graphs 1998-01-13
From Simon Batten:
How does varying the values a,b and c affect the graph of
y=ax2 + bx + c?

Answered by Harley Weston.
(-2)^x 1997-12-12
From Ken Bokinac:
During our math class last week we came accross an equation that we could not graph we were wondering if you could help us try and graph it. The question is:

Answered by Harley Weston.

Graphing Inequalities of Conic Sections 1997-03-24
From James Sheldon:
I'm trying to graph Systems of Conic Sections with inequalities, but I'm running into problems on which area to shade:
x^2+y^2 is greater than or equal to 16
xy > 4

So I graph these two equations, and then my teacher said to substitute a point into it but I'm still not sure how to do it...
Answered by Penny Nom.

Where do 2^x and x^2 intersect? 1996-07-23
From Darin M Selfridge:
The intersecting points of y=2^x and y=x^2, graphed along with x^2=2^x, are obviously three... (2,4), (4,16) and (?,?). I am having problems figuring out the value, and even an equation for the (?,?). Can you help?
Answered by Harley Weston Chris Fisher.
How do you graph these inequalities? 1996-05-17
From Robert:
How do you graph the inequalities 5+gg, and g+h>6 on one coordinate plane?
Answered by Harley Weston and Maxine Stinka.
How do you graph these equations? 1996-04-30
From David Gamble:
How do you graph the equation x=3 on a x,y graph? Also how do you graph f(x)=-3x + 2 on the same graph?
Answered by Harley Weston and Maxine Stinka.
l'équation d'une rotation dans un graphique cartésien 2001-11-14
From Ghaith:
je d6sir que vous me rafraichissez un peu la mémoire s.v.p. j'aimerai savoir l'équation d'une rotation dans un graphique cartésien merci
Answered by Claude Tardif.
la pantographie 1999-11-18
From Erick et Christine:
Nous recherchons la définition de pantographie voltaïque, que représente cette "figure" ? est-ce qu'on peut en définir une unité de mesure ?
Answered by Claude Tardif.



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