QF - Rate of Change (Lesson)

Rate of Change

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

Example: 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.

Example: Calculate the rate of change for the function $f(x)=x^2+10x+4$ 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.

$f(-3)=(-3)^2+10(-3)+4=-17$         -- This gives you the point: (-3, -17)

$f(-1)=(-1)^2+10(-1)+4=-5$            -- This gives you the point: (-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.

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Calculating Rate of Change Practice

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

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

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

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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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.  

 

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Linear, Quadratic, or Neither Functions Practice

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

1.

x	f(x)
-2	-9
-1	-2
0	1
1	0
2	-5

2.

x	f(x)
-2	-16
-1	-8
0	-4
1	-2
2	-1

3.

x	f(x)
-2	17
-1	13
0	9
1	5
2	1

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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