GQF - Rate of Change (Lesson)

Rate of Change

In an earlier module we learned that the rate of change is the slope between two points.  

Given two points, calculate the slope between them:

$m=\frac{11-14}{3-2}=\frac{-3}{1}=-3$ 

If you were to draw a line between the points, the slope of the line would be -3.

If you are not given both the x- and y-parts of the points, you may need to plug in the x-part to find the y-part.

Calculate the rate of change for the function f(x) = x² + 10x + 4 over the interval $-3\le x\le-1$.

You are given the x-parts of each point. So first you should find the y-parts.

f(-3) = (-3)² + 10(-3) + 4 = -17 so (-3, -17)

f(-1) = (-1)² + 10(-1) + 4 = -5 so (-1, -5)

Now calculate the slope between those points

$m=\frac{-5-\left(-17\right)}{-1-\left(-3\right)}=\frac{-5+17}{-1+3}=\frac{12}{2}=6$ 

So the rate of change over that interval is 6.

1. Calculate the rate of change for the function g(x) = 2x² + 8x + 3 over the interval $-2\le x\le0$. 

2. Calculate the rate of change for the function g(x) = -x² + 4x + 2 over the interval $3\le x\le4$. 

3. Calculate the rate of change for the function g(x) = (1/2)x² + 5x - 4 over the interval $-4\le x\le1$. 

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For a linear function, the slope is always constant, but for a quadratic function, the rate of change is constant for the 2^nd difference. Let's watch this video to figure out what that means:

When comparing the rates of change for linear functions to quadratic functions, we can say that a quadratic function has a greater rate of change, because it grows faster than a line.  

Linear, Quadratic, or Neither Functions Practice

Determine if each table below represents a linear function, quadratic function, or neither.

1.

2.

3.

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