Increasing and Decreasing Functions

Question

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = (−x)²/³

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to analyze and gravity function Y is equal to the quantity of minus 8X in quantity rates to the 2/3 power. What symmetries does it exhibit, and during what intervals is it increasing or decreasing? Note Treat the function in its radical form. Below the problem we're given a graph on which to plot our function. Now, the first thing we want to do before we begin to analyze our function is to actually rewrite our function and to simplify it. So if we look at Y as equal to quantity of -8X in quantity, raised to the 2/3 power. We can actually square everything inside the parentheses, but this is going to be equal to the quantity of 64, multiplied by X2 in quantity raised to the 1/3 power. And now taking the cube root of everything inside the parentheses, this is going to be equal to. 4 multiplied by X raised to the 2/3 power. So we're actually going to use this simplified form of our expression for Y in our analysis. So the first thing we want to do is determine the symmetries. And so to do that, to check for even or odd symmetries, we'll calculate Y of minus X. And this is going to be equal to 4, multiplied by the quantity of minus X in quantity raised to the 2/3 power. And this is equal to 4 multiplied by X raised to the 2/3 power. Since when we square the negative sign, it goes away. And this is actually equal to Y of X. So, since we have Y minus X equal to Y of X, that means we have even symmetry. And so our function is going to be symmetric about the Y axis. Now, the next thing we want to do is determine the intervals over which our function is increasing or decreasing. And to do that, we first want to notice that our function Y is always going to be greater than or equal to 0. Since the quantity of X-rays to the 2/3 power is always going to be positive or equal to 0. And so that means we're gonna have a minimum occurring. When Y is equal to 0, and that occurs when X is equal to 0. So, we have a minimum of Y equal to 0 at X equal to 0. And that's the only minimum that we actually have. And so we can use that minimum to sort of split up our function, our domain here, determine the intervals over which our function is increasing or decreasing. So, if we look at the case, Of X greater than 0. That means that our function Y is going to be increasing since we're to the right of our minimum value. Recall that our function increases to the right of a minimum, and so that means our function Y is increasing on the open interval going from 0 to infinity. And if we look at the values of X that are less than 0, we see that Y is going to be decreasing, so Y is decreasing. On the interval, going from minus infinity to 0. So, we call that our function decreases to the values of X that are just to the left of a minimum. So these are two intervals of increasing and decreasing for our function. So the last thing we need to do is just use this information to draw our function. And before we do that, we need to determine a couple of points to plot on a graph. So, we're gonna look at the point of when X is equal to 1, so we'll calculate Y of 1, and this is going to be equal to. 4 multiplied by 1 raised to the 2/3 power which when we evaluate this expression is equal to 4. And by symmetry, Y minus 1 is equal to Y of 1. Which is also going to be equal to 4. So next we'll look at it at an X value of 8. We'll set X equal to 8, and so we'll calculate Y of 8. And this is going to be equal to 4, multiplied by 8, raised to the 2/3 power. And evaluate this expression, this is going to be equal to 16. And again, by symmetry, Y minus 8, which is also equal to Y of 8, is also going to be equal to 16. So we now have 5 points if we include our minimum point at the origin to plot on our graph. So plotting those points, we're starting with the minimum at the origin. Next, we'll look at when X is equal to 1, our Y value is going to be equal to 4, and when X is equal to 8, our Y value is going to be equal to 16. And by symmetry, when X is equal to -1, or Y value is 4, and when X is equal to -8, or Y value is 16. So the last thing we need to look at is how our function Y behaves as X approaches plus or minus infinity. And if we look at X approaching infinity, that means that our function Y is also going to approach infinity, since the quantity of X rays to the 2/3 also approaches infinity as X approaches infinity. Likewise, when X approaches minus infinity, our y value is again going to approach infinity. Since we're squaring the value of X, it's the quantity of X raises to the 2/3 is always going to be positive. And so, therefore, since that is also approaching infinity, our Y value is approaching infinity. So taking all that into account, We're going to draw our function for Y and recall that for the values of X that are greater than 0, our function Y is going to be increasing, so I'll draw an increasing function, connecting our points. That we have plotted on the graph. And likewise, our function is going to be decreasing. When our values of X are negative, they connect our points to the left of X equal to 0 with a curve that is constantly decreased until it reaches X equal to 0. So the graph that we have drawn on the screen is the answer to this question. So I hope that this explanation has been useful, and I'll see you in the next video.

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = (−x)²/³

Key Concepts

Increasing and Decreasing Functions

Graphing Functions

Symmetry in Functions

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