Find the Equation Using Two Points f(-3)=1 and f(13)=5

$f (- 3) = 1$ and $f (13) = 5$

Step 1

$f (- 3) = 1$ , which means $(- 3, 1)$ is a point on the line. $f (13) = 5$ , which means $(13, 5)$ is a point on the line, too.

$(- 3, 1), (13, 5)$

Step 2

Find the slope of the line between $(- 3, 1)$ and $(13, 5)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{5 - (1)}{13 - (- 3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $1$ .

$m = \frac{5 - 1}{13 - (- 3)}$

Subtract $1$ from $5$ .

$m = \frac{4}{13 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{4}{13 + 3}$

Add $13$ and $3$ .

$m = \frac{4}{16}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$m = \frac{4 (1)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

$m = \frac{4 \cdot 1}{4 \cdot 4}$

Cancel the common factor.

$m = \frac{4 \cdot 1}{4 \cdot 4}$

Rewrite the expression.

$m = \frac{1}{4}$

Step 3

Use the slope $\frac{1}{4}$ and a given point $(- 3, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{1}{4} \cdot (x - (- 3))$

Step 4

Simplify the equation and keep it in point-slope form.

$y - 1 = \frac{1}{4} \cdot (x + 3)$

Step 5

Solve for $y$ .

Simplify $\frac{1}{4} \cdot (x + 3)$ .

Rewrite.

$y - 1 = 0 + 0 + \frac{1}{4} \cdot (x + 3)$

Simplify by adding zeros.

$y - 1 = \frac{1}{4} \cdot (x + 3)$

Apply the distributive property.

$y - 1 = \frac{1}{4} x + \frac{1}{4} \cdot 3$

Combine $\frac{1}{4}$ and $x$ .

$y - 1 = \frac{x}{4} + \frac{1}{4} \cdot 3$

Combine $\frac{1}{4}$ and $3$ .

$y - 1 = \frac{x}{4} + \frac{3}{4}$

Move all terms not containing $y$ to the right side of the equation.

Add $1$ to both sides of the equation.

$y = \frac{x}{4} + \frac{3}{4} + 1$

Write $1$ as a fraction with a common denominator.

$y = \frac{x}{4} + \frac{3}{4} + \frac{4}{4}$

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{3 + 4}{4}$

Add $3$ and $4$ .

$y = \frac{x}{4} + \frac{7}{4}$

Step 6

The final answer is the equation in slope-intercept form.

$y = \frac{1}{4} x + \frac{7}{4}$

Step 7

Replace $y$ by $f (x)$ .

$f (x) = \frac{1}{4} x + \frac{7}{4}$

Step 8

Find the Equation Using Two Points f(-3)=1 and f(13)=5

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