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

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

Step 1

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

$(2, - 2), (1, 1)$

Step 2

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

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

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

Add $1$ and $2$ .

$m = \frac{3}{1 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Divide $3$ by $- 1$ .

$m = - 3$

Step 3

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

$y - (- 2) = - 3 \cdot (x - (2))$

Step 4

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

$y + 2 = - 3 \cdot (x - 2)$

Step 5

Solve for $y$ .

Simplify $- 3 \cdot (x - 2)$ .

Rewrite.

$y + 2 = 0 + 0 - 3 \cdot (x - 2)$

Simplify by adding zeros.

$y + 2 = - 3 \cdot (x - 2)$

Apply the distributive property.

$y + 2 = - 3 x - 3 \cdot - 2$

Multiply $- 3$ by $- 2$ .

$y + 2 = - 3 x + 6$

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

Subtract $2$ from both sides of the equation.

$y = - 3 x + 6 - 2$

Subtract $2$ from $6$ .

$y = - 3 x + 4$

Step 6

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

$f (x) = - 3 x + 4$

Step 7

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

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