Find the x and y Intercepts f(x)=-1/8(2x-6)^3-2

$f (x) = - \frac{1}{8} {(2 x - 6)}^{3} - 2$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{1}{8} \cdot {(2 x - 6)}^{3} - 2$

Solve the equation.

Rewrite the equation as $- \frac{1}{8} \cdot {(2 x - 6)}^{3} - 2 = 0$ .

$- \frac{1}{8} \cdot {(2 x - 6)}^{3} - 2 = 0$

Add $2$ to both sides of the equation.

$- \frac{1}{8} \cdot {(2 x - 6)}^{3} = 2$

Simplify $- \frac{1}{8} \cdot {(2 x - 6)}^{3}$ .

Combine ${(2 x - 6)}^{3}$ and $\frac{1}{8}$ .

$- \frac{{(2 x - 6)}^{3}}{8} = 2$

Simplify the numerator.

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

Factor $2$ out of $2 x$ .

$- \frac{{(2 (x) - 6)}^{3}}{8} = 2$

Factor $2$ out of $- 6$ .

$- \frac{{(2 x + 2 \cdot - 3)}^{3}}{8} = 2$

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

$- \frac{{(2 (x - 3))}^{3}}{8} = 2$

Apply the product rule to $2 (x - 3)$ .

$- \frac{2^{3} {(x - 3)}^{3}}{8} = 2$

Raise $2$ to the power of $3$ .

$- \frac{8 {(x - 3)}^{3}}{8} = 2$

Cancel the common factor of $8$ .

Cancel the common factor.

$- \frac{8 {(x - 3)}^{3}}{8} = 2$

Divide ${(x - 3)}^{3}$ by $1$ .

$- {(x - 3)}^{3} = 2$

Divide each term in $- {(x - 3)}^{3} = 2$ by $- 1$ and simplify.

Divide each term in $- {(x - 3)}^{3} = 2$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide ${(x - 3)}^{3}$ by $1$ .

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

Simplify the right side.

Divide $2$ by $- 1$ .

${(x - 3)}^{3} = - 2$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$x - 3 = \sqrt[3]{- 2}$

Simplify $\sqrt[3]{- 2}$ .

Rewrite $- 2$ as ${(- 1)}^{3} \cdot 2$ .

Rewrite $- 2$ as $- 1 (2)$ .

$x - 3 = \sqrt[3]{- 1 (2)}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x - 3 = \sqrt[3]{{(- 1)}^{3} \cdot 2}$

Pull terms out from under the radical.

$x - 3 = - 1 \sqrt[3]{2}$

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$x - 3 = - \sqrt[3]{2}$

Add $3$ to both sides of the equation.

$x = - \sqrt[3]{2} + 3$

x-intercept(s) in point form.

x-intercept(s): $(- \sqrt[3]{2} + 3, 0)$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{1}{8} \cdot {(2 (0) - 6)}^{3} - 2$

Solve the equation.

Remove parentheses.

$y = - \frac{1}{8} \cdot {(2 (0) - 6)}^{3} - 2$

Simplify $- \frac{1}{8} \cdot {(2 (0) - 6)}^{3} - 2$ .

Simplify each term.

Multiply $2$ by $0$ .

$y = - \frac{1}{8} \cdot {(0 - 6)}^{3} - 2$

Subtract $6$ from $0$ .

$y = - \frac{1}{8} \cdot {(- 6)}^{3} - 2$

Raise $- 6$ to the power of $3$ .

$y = - \frac{1}{8} \cdot - 216 - 2$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$y = \frac{- 1}{8} \cdot - 216 - 2$

Factor $8$ out of $- 216$ .

$y = \frac{- 1}{8} \cdot (8 (- 27)) - 2$

Cancel the common factor.

$y = \frac{- 1}{8} \cdot (8 \cdot - 27) - 2$

Rewrite the expression.

$y = - 1 \cdot - 27 - 2$

Multiply $- 1$ by $- 27$ .

$y = 27 - 2$

Subtract $2$ from $27$ .

$y = 25$

y-intercept(s) in point form.

y-intercept(s): $(0, 25)$

List the intersections.

x-intercept(s): $(- \sqrt[3]{2} + 3, 0)$

y-intercept(s): $(0, 25)$

Find the x and y Intercepts f(x)=-1/8(2x-6)^3-2

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