Find the Equation Using Two Points f(-9)=10 and f(-1)=-2

$f (- 9) = 10$ and $f (- 1) = - 2$

$f (- 9) = 10$ , which means $(- 9, 10)$ is a point on the line. $f (- 1) = - 2$ , which means $(- 1, - 2)$ is a point on the line, too.

$(- 9, 10), (- 1, - 2)$

Find the slope of the line between $(- 9, 10)$ and $(- 1, - 2)$ 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.

$m = \frac{- 2 - (10)}{- 1 - (- 9)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $10$ .

$m = \frac{- 2 - 10}{- 1 - (- 9)}$

Subtract $10$ from $- 2$ .

$m = \frac{- 12}{- 1 - (- 9)}$

Simplify the denominator.

Multiply $- 1$ by $- 9$ .

$m = \frac{- 12}{- 1 + 9}$

Add $- 1$ and $9$ .

$m = \frac{- 12}{8}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 12$ and $8$ .

Factor $4$ out of $- 12$ .

$m = \frac{4 (- 3)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$m = \frac{4 \cdot - 3}{4 \cdot 2}$

Cancel the common factor.

$m = \frac{4 \cdot - 3}{4 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Use the slope $- \frac{3}{2}$ and a given point $(- 9, 10)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (10) = - \frac{3}{2} \cdot (x - (- 9))$

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

$y - 10 = - \frac{3}{2} \cdot (x + 9)$

Solve for $y$ .

Simplify $- \frac{3}{2} \cdot (x + 9)$ .

Rewrite.

$y - 10 = 0 + 0 - \frac{3}{2} \cdot (x + 9)$

Simplify by adding zeros.

$y - 10 = - \frac{3}{2} \cdot (x + 9)$

Apply the distributive property.

$y - 10 = - \frac{3}{2} x - \frac{3}{2} \cdot 9$

Combine $x$ and $\frac{3}{2}$ .

$y - 10 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot 9$

Multiply $- \frac{3}{2} \cdot 9$ .

Multiply $9$ by $- 1$ .

$y - 10 = - \frac{x \cdot 3}{2} - 9 (\frac{3}{2})$

Combine $- 9$ and $\frac{3}{2}$ .

$y - 10 = - \frac{x \cdot 3}{2} + \frac{- 9 \cdot 3}{2}$

Multiply $- 9$ by $3$ .

$y - 10 = - \frac{x \cdot 3}{2} + \frac{- 27}{2}$

Simplify each term.

Move $3$ to the left of $x$ .

$y - 10 = - \frac{3 \cdot x}{2} + \frac{- 27}{2}$

Move the negative in front of the fraction.

$y - 10 = - \frac{3 x}{2} - \frac{27}{2}$

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

Add $10$ to both sides of the equation.

$y = - \frac{3 x}{2} - \frac{27}{2} + 10$

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{2} - \frac{27}{2} + 10 \cdot \frac{2}{2}$

Combine $10$ and $\frac{2}{2}$ .

$y = - \frac{3 x}{2} - \frac{27}{2} + \frac{10 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{- 27 + 10 \cdot 2}{2}$

Simplify the numerator.

Multiply $10$ by $2$ .

$y = - \frac{3 x}{2} + \frac{- 27 + 20}{2}$

Add $- 27$ and $20$ .

$y = - \frac{3 x}{2} + \frac{- 7}{2}$

Move the negative in front of the fraction.

$y = - \frac{3 x}{2} - \frac{7}{2}$

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

$y = - \frac{3}{2} x - \frac{7}{2}$

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

$f (x) = - \frac{3}{2} x - \frac{7}{2}$

Find the Equation Using Two Points f(-9)=10 and f(-1)=-2

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