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 = x³/8

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to analyze and graph the function Y is equal to X cube divided by 343. What symmetries does it exhibit and during which intervals is it increasing or decreasing? The the problem where we have an empty graph on which to plot our function. So the first thing we want to look at are the symmetries for our function. And so here we're going to look at Y minus X to check for even our odd symmetry. And so, Y minus X is going to be equal to the quantity of minus X in quantity cubed, divided by 343. Which is equal to minus X cubed divided by 343. Which is equal to minus YFX. Now, since Y minus X is equal to minus Y of X, that means that we have odd symmetry. And so our function here is symmetric about the origin. That's the first part of our question answered. Next, we need to determine the intervals at which our function is increasing or decreasing. And to do this, we're going to need to analyze the first derivative. So we need to calculate Y and that's going to be equal to the derivative with respect to X. Of the quantity of X cubed divided by 343. And here we're gonna have to use our power rule and constant multiple rule to take this derivative. In doing so, this is going to be equal to 3 x 2. Divided by 343. Now, if we look at our function here for the first derivative, we say that it's going to be positive for all x values, except for when X is equal to 0, in which case our first derivative would be equal to 0. And since our first derivative um being positive indicates that our function is increasing, that means that our function in this case, Y, is going to be increasing. Over the intervals Well, over the open interval from minus infinity to 0, union with the open interval from 0 to infinity. So it's increasing for all values of X except for x equal to 0, in which case we'd have a slope of 0. So now to graph our function, we just need to identify some key points. And so we're going to start off by looking at X equal to 0. We'll have Y of 0. This is going to be equal to 0 cubed divided by 343, which is going to be equal to 0. Another point that we're gonna look at is when X is equal to 7, so we'll have Y of 7, and that's going to be equal to 7 cubed, divided by 343. And when we evaluate this expression, this is going to be equal to 1. And we're going to want to make use of our odd symmetry, so we're gonna need to look at X equal to -7, and so we'll have Y of -7. And this is going to be equal to the quantity of minus 7 in quantity cubed, divided by 343. Which is equal to -1. So now we're gonna plot these points on our graph. So we'll plot the point at 0.0. We'll plot the next point at 7.1. And the last point at -7. 1. And we're going to do is connect these points with a smooth curve. Recalling the fact that we need to have our function constantly increasing. So, our function is gonna be increasing? And it's going to level off until it has a slope of 0 at the origin, and then our slope is going to continue to increase. And our graph will continue to increase in its Y value. And so, the graph that we have drawn on the screen is part of the answer for this problem. So just to recap of what we also found, we found that we had odd symmetry, which means that our function is symmetric about the origin, and we also found that our function was increasing over the open interval, going from minus infinity to 0, union with the open interval from 0 to infinity. So those are the answers for this problem. 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 = x³/8

Key Concepts

Increasing and Decreasing Functions

Symmetry in Graphs

Derivative and Critical Points

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