Graph g(x)=-1/3f(x-2)+4

$g (x) = - \frac{1}{3} f (x - 2) + 4$

Step 1

Rewrite the function as an equation.

$y = - \frac{1}{3} \cdot (f (x - 2)) + 4$

Step 2

Remove parentheses.

$y = - \frac{1}{3} \cdot (f (x - 2)) + 4$

Step 3

Simplify $- \frac{1}{3} \cdot (f (x - 2)) + 4$ .

Simplify each term.

Apply the distributive property.

$y = - \frac{1}{3} \cdot (f x + f \cdot - 2) + 4$

Move $- 2$ to the left of $f$ .

$y = - \frac{1}{3} \cdot (f x - 2 \cdot f) + 4$

Apply the distributive property.

$y = - \frac{1}{3} (f x) - \frac{1}{3} (- 2 f) + 4$

Multiply $- \frac{1}{3} (f x)$ .

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

$y = - \frac{f}{3} x - \frac{1}{3} (- 2 f) + 4$

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

$y = - \frac{x f}{3} - \frac{1}{3} (- 2 f) + 4$

Multiply $- \frac{1}{3} (- 2 f)$ .

Multiply $- 2$ by $- 1$ .

$y = - \frac{x f}{3} + 2 (\frac{1}{3}) f + 4$

Combine $2$ and $\frac{1}{3}$ .

$y = - \frac{x f}{3} + \frac{2}{3} f + 4$

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

$y = - \frac{x f}{3} + \frac{2 f}{3} + 4$

Reorder $x$ and $f$ .

$y = - \frac{f x}{3} + \frac{2 f}{3} + 4$

Step 4

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Combine $\frac{2 f}{3} + 4$ .

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{f x}{3} + \frac{2 f}{3} + 4 \cdot \frac{3}{3}$

Simplify terms.

Combine $4$ and $\frac{3}{3}$ .

$y = - \frac{f x}{3} + \frac{2 f}{3} + \frac{4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{f x}{3} + \frac{2 f + 4 \cdot 3}{3}$

Simplify the numerator.

Factor $2$ out of $2 f + 4 \cdot 3$ .

Factor $2$ out of $2 f$ .

$y = - \frac{f x}{3} + \frac{2 (f) + 4 \cdot 3}{3}$

Factor $2$ out of $4 \cdot 3$ .

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

Factor $2$ out of $2 (f) + 2 (2 \cdot 3)$ .

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

Multiply $2$ by $3$ .

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

Rewrite in slope-intercept form.

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

Step 5

The slope and y-intercept cannot be found for this problem since it is not linear.

Not Linear

Step 6

Use the slope-intercept form to find the slope and y-intercept.

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m =$

$b =$

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope:

y-intercept: $(0,)$

Slope:

y-intercept: $(0,)$

Step 7

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & 4 + \frac{2 f}{3} \\ 1 & 4 + \frac{f}{3} \end{array}$

Step 8

Graph the line using the slope and the y-intercept, or the points.

Slope:

y-intercept: $(0,)$

$\begin{array}{c} x & y \\ 0 & 4 + \frac{2 f}{3} \\ 1 & 4 + \frac{f}{3} \end{array}$

Step 9

Graph g(x)=-1/3f(x-2)+4

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