a-sketch-the-graph-of-yx2-b-sketch-the-graph-of-y3-x2-c-sketch-the-graph-of-yfrac13-x2-d-describe-th

Question

a. Sketch the graph of $y=x^{2}$
b. Sketch the graph of $y=3 x^{2}$
c. Sketch the graph of $y=\frac{1}{3} x^{2}$
d. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $a > 1$
e. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $0 < a < 1$

Finian Lickona · Accepted Answer

all right. So for the first part of this exercise, we need to sketch the graph of Why equals X squared. And I&#x27;ve already loaded that just for convenience. And for the second part of the question, we need to sketch the graph of y equals three x squared. So if I plug in zero for X, I get zero. That&#x27;s our first point. If I plug in one for X, I get three times one, which is going to be three. If I plug in negative one, I also get three. So that parable is going to look something like this, and there&#x27;s no curvature down there. Even though there should be. It&#x27;s just kind of hard to draw. Uh, And so, for the next part of the question, we have to sketch the graph of Y equals 1/3 x squared. So that&#x27;s gonna look something like this again. We have zero point. And then if I plug in, let&#x27;s say let&#x27;s say I plug in three for X Squared Perec&#x27;s So I get nine as X squared and 1/3 of nine ist three. So if I go over to three and I have to go up to three, and then the same thing applies for negative three. So this per Abalo looks something like this. Okay, So for the next part of the question, we have to describe the graph of why equals Yeah, why equals a X squared when a is greater than one? So when a is greater than one, the graph of y equals X squared is going to stretch The parabola of the basic graph of y equals X squared. It&#x27;s going to make it longer in skinnier. And then when a is between zero and one, then it compresses the graph. It sort of squishes it and makes it fatter. So we have, ah, bigger paro Ebola or sort of wider problem for smaller values of a between zero and one. And we have a longer, skinnier problem for values of a greater than one

Finian Lickona · Answer

Okay, So for this exercise, the first thing we have to do is sketch the graph of why equals X squared. And I&#x27;ve already pre loaded this graph just for convenience because it helps us expedite the problem of it. So the second part of the question asks us to sketch the graph of y equals X plus two squared. And this is actually going to look exactly like this graph. Except if we plug in negative two for X, why become zero? So that we know are X intercept is going to be at the negative to mark. So I&#x27;m just gonna go ahead and graph in a few more points so we can sort of get an idea of what this parable looks like. Okay. And then, for the third part of the problem, we have to sketch the graph of y equals X minus three. And this is gonna be the same concept as up here. Except if we plug in positive three for X. That is where our X intercept is. That&#x27;s what makes why equal to zero. So this graph is going to be over here. Apologies for the terrible problems, and then we need to describe the graph of y equals X plus a squared in terms of the graph of y equals X squared. So whenever we have this a term, it is going to shift the graft to the left or right, depending on whether the sign of the eight terms negative or positive, we noticed that when the A term has a positive sign, it shifts to the left into the negative Derek option. And when it has a negative sign, it ships to write in a positive direction. So the transformation that maps y equals X squared to y equals X plus a squared is a shift to the left or right, um, by a factor of a So if this is a over here, this would be the graph of why equals X plus. Hey, we&#x27;re gonna say that&#x27;s negative. That would be the graft. A r y equals X will say square right there

Finian Lickona · Answer

So for this exercise, we&#x27;re going to have to graft the functions. Why equals the absolute value of X Y equals two times the absolute value of X, and why equals 1/2 times the absolute value of X. Okay, so the first thing we&#x27;re gonna do is sketching the graph of y equals the absolute value of X. So that&#x27;s pretty straightforward. That is just this graph right here and then because it&#x27;s an absolute value function, all the negative X values generate positive. Weil puts. Okay, so now for this next function, what I&#x27;m gonna do is I&#x27;m gonna plug in one and negative one for the value of X and see what I get is my output. So if I plug it in one, I have two times. One is two. So that means that my output actually puts me right here by program negative one. I get two times the absolute value of negative one, which is also positive too. So I have another point right there. So the graph of y equals two times the absolute value of X actually looks. Yeah, on the 1st 1 Wasn&#x27;t that great? Try that back in real quick all right. Anyway, sorry for the terrible graft, but you get the idea. Okay? And now if I plug in, I&#x27;m gonna pick the values to And negative too, for this next function. And I&#x27;m going to write Ah, 1/2 times. The absolute value, too is one. So I&#x27;m going to go over to the X value to the Y value of one and 1/2 times. They absolutely negative too is also one. So this graph actually looks like this. All right, So now we need Thio are going to want to explore how, in general the function y equals a whereas any constant times the absolute value of exchanges. The graph of y equals the absolute value of X. So this a coefficient is basically a scaling factor. It increases the slope of the line or decreases the slope of the line based on whether or not this a value is less than greater than or equal to one. So if I were to substitute and say five for this, a value that would change this slope of my absolute value function to five. So I have to go over warning up to, like, five, which is all the way up here at anyways. So in general this a value in front of the absolute value term is going to be a scaling coefficient.

Mary Brese · Answer

we&#x27;re looking at, um f of X equals absolute value of X. And when you change the equation, toe f of X equals absolute value of X Plus two or F of X equals absolute value of X minus two. Since they are all absolute value functions, we call this a family of functions. The question is, is what happens when we put a plus two after the original parent function or a minus two? So let&#x27;s take a look at some graphs. We want to look at the absolute value of the absolute value of X first. And so here&#x27;s our absolute value of X. It starts at the origin, and we have a slope of one to the right. We have a absolute value of slope of negative one to the left s. So this is your ab, your parent function of absolute value of X. But what happens if we add to to the right hand side? So we&#x27;ve got absolute value of X and on the outside, we&#x27;re going to add to, so if you&#x27;ll notice, we still have a slope of one, and we know it&#x27;s a slope of one without even looking at the grid because our blue graph are apparent, function is parallel to our new function. The only thing is the Vertex point shifted up one. So for each x value, the graph shifted up. I mean shifted up to because in our parent function, when X was one, why is also one? But now on X is one. Why is three when X is too? Why was to now exes? Four. So you can see that adding to outside the absolute value bars. We call this a vertical shift and you&#x27;re adding to toe all of the original function values. So that shifts up to what happens if we put a minus two. You probably can guess at this point let&#x27;s do the I&#x27;m suit value of X. And this time we&#x27;re going. Teoh, subtract two and you&#x27;ll see that our third function again. Our vertex point is when x zero, because that hasn&#x27;t changed. So the only thing that changed is once we found our original function value all of those values we subtracted to from all of the original by values so adding or subtracting a constant number outside of your absolute value bars is called a vertical shift So let&#x27;s go back to our equations. This is our parent function. The Vertex is at 00 and this is a vertical shift. I&#x27;m sorry. This is a vertical shift up. One are up to, and this is a vertical shift down to

Finian Lickona · Answer

All right, So in this exercise, we are asked to draw a couple of graphs and then compare the two. So the first thing we&#x27;re going to have to do is try The graph of y equals the absolute value of X. So if we were to draw that, it would look something like this. We have the equation y equals X for all positive values of X. And we have the equation. Why equals negative X for all negative values of X? So we get this kind of symmetric V about the origin. Because if we were to say plug in negative five, we get y equals the absolute value negative or let&#x27;s do negative for which is positive for so the y value would be positive for Okay, so we&#x27;re just gonna un draw that graph. And the second graf we have to sketch is the graph of y equals the absolute value of X plus two. So now this is going to look exactly like the other graph that we just drew, except that it&#x27;s going to have a shift to it. So if we were to plug in the value of negative two for X a negative to pose positive to zero. So that would be what makes y equals zero. So we have the exact same sort of V shaped with the exact same slope on either side, except it&#x27;s been shifted over to the negative to spot. And now we need to sketch the graph of why equals the absolute value of X minus two. Now, this is the exact same thing. Only instead of shifting over to the left, we&#x27;re gonna be shifting over to the right because if we plug in positive two for X, that&#x27;s what makes White with zero. So that&#x27;s that graph right there. And okay, now we need thio generally describe really to generally described the graph of y equals X plus a where a is any constant in terms of the graph of y equals x. So as we conceive from the previous example, what adding this a term does is it shifts the graph to the left or right, depending on the sign of the constant by whatever the value of A is. So if I were to say that a is equal to negative three, for example, I shift my graph so that its Vertex is real action on a equals positive three. Then I shift my graph over to the left in the negative direction because it takes a negative value to cancel out a positive value. All right, so in general, the, uh, constant is going to shift the graft to the left or the right. If it&#x27;s inside the absolute value brackets, it&#x27;s not gonna change this slope. And it&#x27;s also not going to change the vertical.

Abigail Martyr · Answer

and this problem. First, we are graphing the function each negative x squared plus y squared. And here I&#x27;ve already drafted and this is what the graph looks like in part B. We were discussing how the release to function out of function. Jiri leads a function f if g is e to the negative a times the quantity x squared plus y squared values of a gridded and zero. And what happens as we as a increases England? What we see happen in this first, our work check there is just a maximum when our graph actually decreases. So in the original graph of part eight, Vertex is right where these equal toe want. But as a gets larger, that maximum drops to below one. And we also see that the graph becomes a little more stick. So like looking at a hill that becoming steeper and steeper, steeper and steeper. So, um, I will say that the slow increases or becomes deeper

Problem

For the parabola whose equation is $y=a x^{2}+b x…

Problem 24 Hard Difficulty

a. Sketch the graph of $y=x^{2}$
b. Sketch the graph of $y=3 x^{2}$
c. Sketch the graph of $y=\frac{1}{3} x^{2}$
d. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $a > 1$
e. Describe the graph of $y=a x^{2}$ in terms of the graph of $y=x^{2}$ when $0 < a < 1$

Answer

See the graph
d) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$
e) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$

Related Courses

Algebra 2 and Trigonometry

Related Topics

Discussion

Top Algebra Educators

Catherine R.

Heather Z.

Alayna H.

Kristen K.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

Ann Xavier Gantert

Algebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

See the graphd) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$e) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$

Algebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Catherine R.

Heather Z.

Alayna H.

Kristen K.

See the graph
d) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$
e) We notice that the graph $y=a x^{2}$ is stretched horizontally by factor $a$

Ann Xavier Gantert

Algebra 2 and Trigonometry