Precalculus Examples

$(1, f (1))$ , $(2, f (2))$

Step 1

Find the slope of the line between $(1, f (1))$ and $(2, f (2))$ 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 1.1

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 1.2

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 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $f$ out of $f (2) - f \cdot 1$ .

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Step 1.4.1.1.1

Factor $f$ out of $- f \cdot 1$ .

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

Step 1.4.1.1.2

Factor $f$ out of $f (2) + f (- 1 \cdot 1)$ .

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

Step 1.4.1.2

Multiply $- 1$ by $1$ .

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

Step 1.4.1.3

Subtract $1$ from $2$ .

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

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $1$ .

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

Step 1.4.2.2

Subtract $1$ from $2$ .

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

Step 1.4.3

Simplify.

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Step 1.4.3.1

Multiply $f$ by $1$ .

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

Step 1.4.3.2

Divide $f$ by $1$ .

$m = f$

Step 2

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

$y - (f (1)) = f \cdot (x - (1))$

Step 3

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

$y - f = f \cdot (x - 1)$

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by multiplying through.

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Step 4.1.2.1

Apply the distributive property.

$y - f = f x + f \cdot - 1$

Step 4.1.2.2

Move $- 1$ to the left of $f$ .

$y - f = f x - 1 \cdot f$

Step 4.1.3

Rewrite $- 1 f$ as $- f$ .

$y - f = f x - f$

Step 4.2

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

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Step 4.2.1

Add $f$ to both sides of the equation.

$y = f x - f + f$

Step 4.2.2

Combine the opposite terms in $f x - f + f$ .

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Step 4.2.2.1

Add $- f$ and $f$ .

$y = f x + 0$

Step 4.2.2.2

Add $f x$ and $0$ .

$y = f x$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = f x$

Point-slope form:

$y - f = f \cdot (x - 1)$

Step 6