Graphing

In Exercises 69–76,...

Question

Graphing

In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.

y = (−2x)²/³

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says graphically function Y is equal to the quantity of minus 5 X in quantity raised to the 2/3 power. Using transformations on the graph of Y is equal to X raised to the 2/3 power. Below the problem, we're given a graph of our function Y is equal to X raised to the 2/3 power, and this graph has a curved V shape with a vertex at the origin. Below that graph, We have another empty graph on which to plot our function. So we need to plot The function Y is equal to the quantity of minus 5 X in quantity rates to the 2/3 power using transformations. And so we're gonna start off by rewriting our function for Y. So we have Y, which is equal to the quantity of minus 5 X in quantity raised to the 2/3 power. And so here we're going to be squaring the quantity in the parentheses and then taking the cube root. And when we square the quantity in parentheses, the negative sign will actually disappear, since we'll be squaring -1 and -1 squad is equal to 1. That means that our function Y is equal to the quantity of 5 X in quantity raised to the 2/3 power. So, if we compare this function to the graph of Y is equal to X, raised to the 2/3 power, what we're doing is we're rescaling our X-axis. So, we're actually going to squish our X-axis by a factor of 5. So that means we're going to get a graph that's similar to the graph that we have or Y is equal to X-rays to the 2/3 power. However, we're going to have a much steeper slope around the origin. So we're gonna need to look at a couple of points on our graph Y is equal to X rates to the 2/3 power and transform those points onto our new graph for Y is equal to the quantity of minus 5 X in quantity rates, so the 2/3 power. And so the first point that we're going to look at is 1.1. And if we need to transform this by dividing the X coordinate by 5, that means that this point is going to get transformed to the point of 1.5.1. And since our function here is symmetric about the Y axis, we also want to have the point of -1.1. And that's going to transform to the point of -1 divided by 5.1. Another point that we can look at is the point of 8.4. On our graph, and this gets transformed to the point of 8 divided by 54. And likewise, by symmetry, we'll have the point of -8.4. On our graph. I want you to transform that point. So this is going to be transformed too. -8 divided by 5.4. Lastly, we'll look at the origin. So, our point of 0.0, on a graph of Y is equal to X-rays to the 2/3 power, that gets transformed to the same point of 0.0. So when I divide 0 by 5, I still get back 0. So we're going to use these points to plot our graph of Y is equal to the quantity of minus 5 X in quantity raised to the 2/3 power. So we'll start off by um graphing our point at the origin. And then the next point we want to look at is going to be 1 divided by 51. Then we'll also need to plot -1 divided by 5.1. The next point we're going to look at is going to be 8 divided by 5.4. We'll plot that point. And then we'll have -8 divided by 5.4 as well, for another point. And so now we want to do is connect these points with a smooth curve. And extended out towards the edges of the graph. And so we'll still have a vertex at the origin. However, The slope of our graph around the origin is much steeper in our new graph of Y is equal to the quantity of minus 5 X in quantity rates of 2/3 power. So the graph that we have drawn on the screen here is the answer to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Graphing

In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.

y = (−2x)²/³

Key Concepts

Standard Functions

Function Transformations

Fractional Exponents

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