# Derivatives Affecting Shape of the Graph

In summary, the conversation discusses how to analyze the function h(x) = (x^2-1)^3, including finding the intervals of increase or decrease, local maximum and minimum values, intervals of concavity and inflection points, and sketching the graph using this information. The first step is to find the derivative of the function, and then to determine how to find the intervals of increase or decrease. The conversation also touches on the concept of a function being increasing and how to determine where a differentiable function is increasing.
I have the following question:

h(x) = (x^2-1)^3

a) Find the intervals of increase or decrease
b) Find the local maximum and minimum values
c) Find the intervals of concavity and the inflection points
d) Use the informatin from parts (a)-(c) to sketch the graph

Now I figure a good way of starting this question is just to graph it to begin with anyways. However, the first step really is to find the derivative (f') of this function if I am not mistaken. However, can anyone explain to me how you would go about finding the intervals of increase or decrease for this function. From there, I can figure out the rest. That's all, thanks guys.

Well, what does it mean that a function is increasing?
And how may you ascertain where a differentiable function is increasing?

## 1. How do derivatives affect the shape of a graph?

Derivatives, also known as the slope or rate of change, can affect the shape of a graph by indicating whether the function is increasing or decreasing at a certain point. If the derivative is positive, the graph will have a positive slope and be increasing. If the derivative is negative, the graph will have a negative slope and be decreasing. The derivative can also indicate the concavity of a graph, with a positive derivative indicating a concave up graph and a negative derivative indicating a concave down graph.

## 2. What is the relationship between derivatives and local extrema?

Local extrema, or the maximum and minimum points of a graph, can be found using derivatives. At these points, the derivative will be equal to zero, indicating a horizontal tangent line. If the derivative changes from positive to negative at this point, it is a local maximum, and if it changes from negative to positive, it is a local minimum. This relationship can help determine the behavior of a graph at these critical points.

## 3. How do derivatives affect the symmetry of a graph?

The symmetry of a graph can be affected by the first and second derivatives. If the first derivative is odd, meaning it is symmetric with respect to the origin, the graph will have a line of symmetry at the origin. If the second derivative is even, meaning it is symmetric with respect to the y-axis, the graph will have a line of symmetry at the y-axis. These symmetries can help identify the shape and behavior of a graph.

## 4. Can derivatives affect the intercepts of a graph?

Derivatives have no effect on the x-intercepts of a graph, as these are determined by the roots of the original function. However, the derivative can affect the y-intercept of a graph. If the original function has a constant term, the derivative will be zero at the y-intercept, indicating a horizontal tangent line. If the derivative is non-zero at the y-intercept, the graph will not have a y-intercept.

## 5. How can derivatives be used to find the average rate of change?

The average rate of change, or the average slope between two points, can be found using derivatives. The derivative of a function at a specific point represents the instantaneous rate of change at that point. By finding the derivative at two different points and taking the average, the average rate of change between those points can be determined. This can be useful in analyzing the behavior of a graph and predicting future trends.

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