Calculus Examples

$f (\frac{1}{2}) = - \frac{5}{3}$ , $f (9) = 4$

Step 1

$f (\frac{1}{2}) = - \frac{5}{3}$ , which means $(\frac{1}{2}, - \frac{5}{3})$ is a point on the line. $f (9) = 4$ , which means $(9, 4)$ is a point on the line, too.

$(\frac{1}{2}, - \frac{5}{3}), (9, 4)$

Step 2

Find the slope of the line between $(\frac{1}{2}, - \frac{5}{3})$ and $(9, 4)$ 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.

$m = \frac{4 - (- \frac{5}{3})}{9 - (\frac{1}{2})}$

Step 2.4

Simplify.

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Step 2.4.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 2.4.1.1

Multiply $\frac{4 - - \frac{5}{3}}{9 - \frac{1}{2}}$ by $\frac{2}{2}$ .

$m = \frac{2}{2} \cdot \frac{4 + \frac{5}{3}}{9 - \frac{1}{2}}$

Step 2.4.1.2

Combine.

$m = \frac{2 (4 + \frac{5}{3})}{2 (9 - \frac{1}{2})}$

Step 2.4.2

Apply the distributive property.

$m = \frac{2 \cdot 4 + 2 (\frac{5}{3})}{2 \cdot 9 + 2 (- \frac{1}{2})}$

Step 2.4.3

Cancel the common factor of $2$ .

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Step 2.4.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$m = \frac{2 \cdot 4 + 2 (\frac{5}{3})}{2 \cdot 9 + 2 (\frac{- 1}{2})}$

Step 2.4.3.2

Cancel the common factor.

$m = \frac{2 \cdot 4 + 2 (\frac{5}{3})}{2 \cdot 9 + 2 (\frac{- 1}{2})}$

Step 2.4.3.3

Rewrite the expression.

$m = \frac{2 \cdot 4 + 2 (\frac{5}{3})}{2 \cdot 9 - 1}$

Step 2.4.4

Simplify the numerator.

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Step 2.4.4.1

Multiply $2$ by $4$ .

$m = \frac{8 + 2 (\frac{5}{3})}{2 \cdot 9 - 1}$

Step 2.4.4.2

Multiply $- - \frac{5}{3}$ .

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Step 2.4.4.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{8 + 2 (1 (\frac{5}{3}))}{2 \cdot 9 - 1}$

Step 2.4.4.2.2

Multiply $\frac{5}{3}$ by $1$ .

$m = \frac{8 + 2 (\frac{5}{3})}{2 \cdot 9 - 1}$

Step 2.4.4.3

Multiply $2 (\frac{5}{3})$ .

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Step 2.4.4.3.1

Combine $2$ and $\frac{5}{3}$ .

$m = \frac{8 + \frac{2 \cdot 5}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.3.2

Multiply $2$ by $5$ .

$m = \frac{8 + \frac{10}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.4

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

$m = \frac{8 \cdot \frac{3}{3} + \frac{10}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.5

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

$m = \frac{\frac{8 \cdot 3}{3} + \frac{10}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.6

Combine the numerators over the common denominator.

$m = \frac{\frac{8 \cdot 3 + 10}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.7

Simplify the numerator.

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Step 2.4.4.7.1

Multiply $8$ by $3$ .

$m = \frac{\frac{24 + 10}{3}}{2 \cdot 9 - 1}$

Step 2.4.4.7.2

Add $24$ and $10$ .

$m = \frac{\frac{34}{3}}{2 \cdot 9 - 1}$

Step 2.4.5

Simplify the denominator.

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Step 2.4.5.1

Multiply $2$ by $9$ .

$m = \frac{\frac{34}{3}}{18 - 1}$

Step 2.4.5.2

Subtract $1$ from $18$ .

$m = \frac{\frac{34}{3}}{17}$

Step 2.4.6

Multiply the numerator by the reciprocal of the denominator.

$m = \frac{34}{3} \cdot \frac{1}{17}$

Step 2.4.7

Cancel the common factor of $17$ .

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Step 2.4.7.1

Factor $17$ out of $34$ .

$m = \frac{17 (2)}{3} \cdot \frac{1}{17}$

Step 2.4.7.2

Cancel the common factor.

$m = \frac{17 \cdot 2}{3} \cdot \frac{1}{17}$

Step 2.4.7.3

Rewrite the expression.

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

Step 3

Use the slope $\frac{2}{3}$ and a given point $(\frac{1}{2}, - \frac{5}{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{5}{3}) = \frac{2}{3} \cdot (x - (\frac{1}{2}))$

Step 4

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

$y + \frac{5}{3} = \frac{2}{3} \cdot (x - \frac{1}{2})$

Step 5

Solve for $y$ .

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Step 5.1

Simplify $\frac{2}{3} \cdot (x - \frac{1}{2})$ .

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Step 5.1.1

Rewrite.

$y + \frac{5}{3} = 0 + 0 + \frac{2}{3} \cdot (x - \frac{1}{2})$

Step 5.1.2

Simplify by adding zeros.

$y + \frac{5}{3} = \frac{2}{3} \cdot (x - \frac{1}{2})$

Step 5.1.3

Apply the distributive property.

$y + \frac{5}{3} = \frac{2}{3} x + \frac{2}{3} (- \frac{1}{2})$

Step 5.1.4

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

$y + \frac{5}{3} = \frac{2 x}{3} + \frac{2}{3} (- \frac{1}{2})$

Step 5.1.5

Cancel the common factor of $2$ .

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Step 5.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y + \frac{5}{3} = \frac{2 x}{3} + \frac{2}{3} \cdot \frac{- 1}{2}$

Step 5.1.5.2

Cancel the common factor.

$y + \frac{5}{3} = \frac{2 x}{3} + \frac{2}{3} \cdot \frac{- 1}{2}$

Step 5.1.5.3

Rewrite the expression.

$y + \frac{5}{3} = \frac{2 x}{3} + \frac{1}{3} \cdot - 1$

Step 5.1.6

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

$y + \frac{5}{3} = \frac{2 x}{3} + \frac{- 1}{3}$

Step 5.1.7

Move the negative in front of the fraction.

$y + \frac{5}{3} = \frac{2 x}{3} - \frac{1}{3}$

Step 5.2

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

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Step 5.2.1

Subtract $\frac{5}{3}$ from both sides of the equation.

$y = \frac{2 x}{3} - \frac{1}{3} - \frac{5}{3}$

Step 5.2.2

Combine the numerators over the common denominator.

$y = \frac{2 x - 1 - 5}{3}$

Step 5.2.3

Subtract $5$ from $- 1$ .

$y = \frac{2 x - 6}{3}$

Step 5.2.4

Factor $2$ out of $2 x - 6$ .

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Step 5.2.4.1

Factor $2$ out of $2 x$ .

$y = \frac{2 (x) - 6}{3}$

Step 5.2.4.2

Factor $2$ out of $- 6$ .

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

Step 5.2.4.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

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

Step 6

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

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

Step 7

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

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

Step 8