Finite Math Examples

$f (1) = 2$ , $f (0) = - 1$

Step 1

$f (1) = 2$ , which means $(1, 2)$ is a point on the line. $f (0) = - 1$ , which means $(0, - 1)$ is a point on the line, too.

$(1, 2), (0, - 1)$

Step 2

Find the slope of the line between $(1, 2)$ and $(0, - 1)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 2.1

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

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

Step 2.2

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).

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

Step 2.3

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

$m = \frac{- 1 - (2)}{0 - (1)}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Multiply $- 1$ by $2$ .

$m = \frac{- 1 - 2}{0 - (1)}$

Step 2.4.1.2

Subtract $2$ from $- 1$ .

$m = \frac{- 3}{0 - (1)}$

Step 2.4.2

Simplify the denominator.

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Step 2.4.2.1

Multiply $- 1$ by $1$ .

$m = \frac{- 3}{0 - 1}$

Step 2.4.2.2

Subtract $1$ from $0$ .

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

Step 2.4.3

Divide $- 3$ by $- 1$ .

$m = 3$

Step 3

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

$y - (2) = 3 \cdot (x - (1))$

Step 4

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

$y - 2 = 3 \cdot (x - 1)$

Step 5

Solve for $y$ .

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Step 5.1

Simplify $3 \cdot (x - 1)$ .

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Step 5.1.1

Rewrite.

$y - 2 = 0 + 0 + 3 \cdot (x - 1)$

Step 5.1.2

Simplify by adding zeros.

$y - 2 = 3 \cdot (x - 1)$

Step 5.1.3

Apply the distributive property.

$y - 2 = 3 x + 3 \cdot - 1$

Step 5.1.4

Multiply $3$ by $- 1$ .

$y - 2 = 3 x - 3$

Step 5.2

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

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Step 5.2.1

Add $2$ to both sides of the equation.

$y = 3 x - 3 + 2$

Step 5.2.2

Add $- 3$ and $2$ .

$y = 3 x - 1$

Step 6

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

$f (x) = 3 x - 1$

Step 7