Precalculus Examples

$f (x) = 4 x^{3} - 2 x^{2} + 9 x + 11$

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 $x$ .

$0 = 4 x^{3} - 2 x^{2} + 9 x + 11$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 4 x^{3} - 2 x^{2} + 9 x + 11$

Step 1.2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.82341982$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 0.82341982, 0)$

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 $y$ .

$y = 4 {(0)}^{3} - 2 {(0)}^{2} + 9 (0) + 11$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{3} - 2 {(0)}^{2} + 9 (0) + 11$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot 0^{3} - 2 \cdot 0^{2} + 9 (0) + 11$

Step 2.2.3

Remove parentheses.

$y = 4 {(0)}^{3} - 2 {(0)}^{2} + 9 (0) + 11$

Step 2.2.4

Simplify $4 {(0)}^{3} - 2 {(0)}^{2} + 9 (0) + 11$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 4 \cdot 0 - 2 {(0)}^{2} + 9 (0) + 11$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 2 {(0)}^{2} + 9 (0) + 11$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 - 2 \cdot 0 + 9 (0) + 11$

Step 2.2.4.1.4

Multiply $- 2$ by $0$ .

$y = 0 + 0 + 9 (0) + 11$

Step 2.2.4.1.5

Multiply $9$ by $0$ .

$y = 0 + 0 + 0 + 11$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 11$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 11$

Step 2.2.4.2.3

Add $0$ and $11$ .

$y = 11$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 11)$

Step 3

List the intersections.

x-intercept(s): $(- 0.82341982, 0)$

y-intercept(s): $(0, 11)$

Step 4