Examples

y = 3 x - 5

$y = 3 x - 5$ ,

(4, 7)

$(4, 7)$ ,

(1, - 2)

$(1, - 2)$

Step 1

Write

y = 3 x - 5

$y = 3 x - 5$ as a function.

f (x) = 3 x - 5

$f (x) = 3 x - 5$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in

y

$y$ values of the two points divided by the change in

x

$x$ values of the two points.

\frac{f (4) - f (1)}{(4) - (1)}

$\frac{f (4) - f (1)}{(4) - (1)}$

Step 2.2

Substitute the equation

y = 3 x - 5

$y = 3 x - 5$ for

f (4)

$f (4)$ and

f (1)

$f (1)$ , replacing

x

$x$ in the function with the corresponding

x

$x$ value.

\frac{(3 (4) - 5) - (3 (1) - 5)}{(4) - (1)}

$\frac{(3 (4) - 5) - (3 (1) - 5)}{(4) - (1)}$

\frac{(3 (4) - 5) - (3 (1) - 5)}{(4) - (1)}

$\frac{(3 (4) - 5) - (3 (1) - 5)}{(4) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply

3

$3$ by

4

$4$ .

\frac{12 - 5 - (3 (1) - 5)}{4 - (1)}

$\frac{12 - 5 - (3 (1) - 5)}{4 - (1)}$

Step 3.1.2

Multiply

3

$3$ by

1

$1$ .

\frac{12 - 5 - (3 - 5)}{4 - (1)}

$\frac{12 - 5 - (3 - 5)}{4 - (1)}$

Step 3.1.3

Subtract

5

$5$ from

3

$3$ .

\frac{12 - 5 - - 2}{4 - (1)}

$\frac{12 - 5 - - 2}{4 - (1)}$

Step 3.1.4

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

\frac{12 - 5 + 2}{4 - (1)}

$\frac{12 - 5 + 2}{4 - (1)}$

Step 3.1.5

Subtract

5

$5$ from

12

$12$ .

\frac{7 + 2}{4 - (1)}

$\frac{7 + 2}{4 - (1)}$

Step 3.1.6

Add

7

$7$ and

2

$2$ .

\frac{9}{4 - (1)}

$\frac{9}{4 - (1)}$

\frac{9}{4 - (1)}

$\frac{9}{4 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{9}{4 - 1}

$\frac{9}{4 - 1}$

Step 3.2.2

Subtract

1

$1$ from

4

$4$ .

\frac{9}{3}

$\frac{9}{3}$

\frac{9}{3}

$\frac{9}{3}$

Step 3.3

Divide

9

$9$ by

3

$3$ .

3

$3$

3

$3$

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