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

Accepted Answer

Welcome back, everyone. In this problem, we want to graph the function Y equals 2x4 and describe its characteristics including symmetries and intervals of increase or decrease. Now, we know that we can graph the function given the empty grid we have on our screen, but we can also think about the, the, the characteristics of our graph as we go along. And then after we're finished, because those will help us to do the graph, we can include the symmetry and intervals of increase or decrease. So let's start by thinking about some points that may be on the graph Y equals 2 X the 4th. Now we know that this function is a polynomial function with an even exponent, and since the exponent is for an even number, that means the graph is U-shaped and thus it's going to be symmetric with respect to the y axis. So, for example, when X equals 0, OK. Y is going to be equal to 2 multiplied by 0 to the 4th, which is 0, and we know that 00 is going to be on our graph, OK. Now, because it's U-shaped and symmetric, then we can figure out if our graph is even, odd, or neither. Now what do we know about graphs that are symmetric about the Y axis? Well, We also know that a function is isn't, OK, that is. Symmetric above the y axis. If OK. F of negative X is equal to F of X. Let's verify that. Notice here that F of negative X. Would be equal to 2 multiplied by negative X the 4th and negative X 4th becomes x 4, which equals 2 x 4, which is the same value as F of X. So this is how we know for sure that our function is even. So therefore, Y is even, OK? Now what else can we test? Well, let's try to understand how our function will behave at the ends. That is where it increases or decreases. Now if we think about it. As X gets larger, OK, as X increases, OK, then. Why will also increase rapidly because the term raise to the 4th X to the 4th dominates and multiplying by 2 makes it grow even faster. The graph never decrease decreases as the values of Y are always positive or zero. So that means then that as X approaches negative infinity, in other words, as X decreases or becomes more negative, then X 4 still increases because even powers make negative numbers positive and thus Y will still be approaching infinity. OK? So no. Since we have our local minimum at 00, the lowest point on our graph, we know it's symmetrical above the y axis. We know the function is decreasing on the interval negative infinity to zero and increasing on the interval 0 to infinity, then we can go ahead and plot our graph, OK? So, if we do that, we know that our graph is going to look something like this. OK. We know that it's gonna bottom out much more quickly than the graph of X squared. So from this we can tell then that. Our function is even. Which means it is symmetric. The metric about the Y axis. Our function decreases. OK, decreases. On the interval. Negative infinity 0. And it increases. On the interval. From 0 to infinity. Thanks a lot for watching everyone. I hope this video helped.

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

Key Concepts

Increasing and Decreasing Functions

Critical Points

Symmetry in Functions

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