Calculus Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = 2 x {(x - 2)}^{3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x {(x - 2)}^{3} = 0$ .

$2 x {(x - 2)}^{3} = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

${(x - 2)}^{3} = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set ${(x - 2)}^{3}$ equal to $0$ .

${(x - 2)}^{3} = 0$

Step 1.2.4.2

Solve ${(x - 2)}^{3} = 0$ for $x$ .

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Step 1.2.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

The final solution is all the values that make $2 x {(x - 2)}^{3} = 0$ true.

$x = 0, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = 2 (0) {((0) - 2)}^{3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 (0) {(0 - 2)}^{3}$

Step 2.2.2

Remove parentheses.

$y = 2 (0) {((0) - 2)}^{3}$

Step 2.2.3

Simplify $2 (0) {((0) - 2)}^{3}$ .

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Step 2.2.3.1

Multiply $2$ by $0$ .

$y = 0 {((0) - 2)}^{3}$

Step 2.2.3.2

Subtract $2$ from $0$ .

$y = 0 {(- 2)}^{3}$

Step 2.2.3.3

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

$y = 0 \cdot - 8$

Step 2.2.3.4

Multiply $0$ by $- 8$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (2, 0)$

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

Step 4