Find the Equation Using Two Points f(-1)=8 f(5)=6

$f (- 1) = 8$ $f (5) = 6$

Step 1

$f (- 1) = 8$ , which means $(- 1, 8)$ is a point on the line. $f (5) = 6$ , which means $(5, 6)$ is a point on the line, too.

$(- 1, 8), (5, 6)$

Step 2

Find the slope of the line between $(- 1, 8)$ and $(5, 6)$ 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.

$m = \frac{6 - (8)}{5 - (- 1)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $8$ .

$m = \frac{6 - 8}{5 - (- 1)}$

Subtract $8$ from $6$ .

$m = \frac{- 2}{5 - (- 1)}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 2}{5 + 1}$

Add $5$ and $1$ .

$m = \frac{- 2}{6}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2$ .

$m = \frac{2 (- 1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$m = \frac{2 \cdot - 1}{2 \cdot 3}$

Cancel the common factor.

$m = \frac{2 \cdot - 1}{2 \cdot 3}$

Rewrite the expression.

$m = \frac{- 1}{3}$

Move the negative in front of the fraction.

$m = - \frac{1}{3}$

Step 3

Use the slope $- \frac{1}{3}$ and a given point $(- 1, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - \frac{1}{3} \cdot (x - (- 1))$

Step 4

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

$y - 8 = - \frac{1}{3} \cdot (x + 1)$

Step 5

Solve for $y$ .

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

Rewrite.

$y - 8 = 0 + 0 - \frac{1}{3} \cdot (x + 1)$

Simplify by adding zeros.

$y - 8 = - \frac{1}{3} \cdot (x + 1)$

Apply the distributive property.

$y - 8 = - \frac{1}{3} x - \frac{1}{3} \cdot 1$

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

$y - 8 = - \frac{x}{3} - \frac{1}{3} \cdot 1$

Multiply $- 1$ by $1$ .

$y - 8 = - \frac{x}{3} - \frac{1}{3}$

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

Add $8$ to both sides of the equation.

$y = - \frac{x}{3} - \frac{1}{3} + 8$

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{3} - \frac{1}{3} + 8 \cdot \frac{3}{3}$

Combine $8$ and $\frac{3}{3}$ .

$y = - \frac{x}{3} - \frac{1}{3} + \frac{8 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{- 1 + 8 \cdot 3}{3}$

Simplify the numerator.

Multiply $8$ by $3$ .

$y = - \frac{x}{3} + \frac{- 1 + 24}{3}$

Add $- 1$ and $24$ .

$y = - \frac{x}{3} + \frac{23}{3}$

Step 6

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

$y = - \frac{1}{3} x + \frac{23}{3}$

Step 7

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

$f (x) = - \frac{1}{3} x + \frac{23}{3}$

Step 8

Find the Equation Using Two Points f(-1)=8 f(5)=6

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