Precalculus Examples

$f (x) = - 4 x^{3} + 4 x^{2} + 15 x$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - 4 {(- 1)}^{3} + 4 {(- 1)}^{2} + 15 (- 1)$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise $- 1$ to the power of $3$ .

$f (- 1) = - 4 \cdot - 1 + 4 {(- 1)}^{2} + 15 (- 1)$

Step 1.2.1.2

Multiply $- 4$ by $- 1$ .

$f (- 1) = 4 + 4 {(- 1)}^{2} + 15 (- 1)$

Step 1.2.1.3

Raise $- 1$ to the power of $2$ .

$f (- 1) = 4 + 4 \cdot 1 + 15 (- 1)$

Step 1.2.1.4

Multiply $4$ by $1$ .

$f (- 1) = 4 + 4 + 15 (- 1)$

Step 1.2.1.5

Multiply $15$ by $- 1$ .

$f (- 1) = 4 + 4 - 15$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $4$ and $4$ .

$f (- 1) = 8 - 15$

Step 1.2.2.2

Subtract $15$ from $8$ .

$f (- 1) = - 7$

Step 1.2.3

The final answer is $- 7$ .

$- 7$

Step 1.3

Convert $- 7$ to decimal.

$y = - 7$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 4 {(0)}^{3} + 4 {(0)}^{2} + 15 (0)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (0) = - 4 \cdot 0 + 4 {(0)}^{2} + 15 (0)$

Step 2.2.1.2

Multiply $- 4$ by $0$ .

$f (0) = 0 + 4 {(0)}^{2} + 15 (0)$

Step 2.2.1.3

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

$f (0) = 0 + 4 \cdot 0 + 15 (0)$

Step 2.2.1.4

Multiply $4$ by $0$ .

$f (0) = 0 + 0 + 15 (0)$

Step 2.2.1.5

Multiply $15$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - 4 {(1)}^{3} + 4 {(1)}^{2} + 15 (1)$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

One to any power is one.

$f (1) = - 4 \cdot 1 + 4 {(1)}^{2} + 15 (1)$

Step 3.2.1.2

Multiply $- 4$ by $1$ .

$f (1) = - 4 + 4 {(1)}^{2} + 15 (1)$

Step 3.2.1.3

One to any power is one.

$f (1) = - 4 + 4 \cdot 1 + 15 (1)$

Step 3.2.1.4

Multiply $4$ by $1$ .

$f (1) = - 4 + 4 + 15 (1)$

Step 3.2.1.5

Multiply $15$ by $1$ .

$f (1) = - 4 + 4 + 15$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 4$ and $4$ .

$f (1) = 0 + 15$

Step 3.2.2.2

Add $0$ and $15$ .

$f (1) = 15$

Step 3.2.3

The final answer is $15$ .

$15$

Step 3.3

Convert $15$ to decimal.

$y = 15$

Step 4

Find the point at $x = 2$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - 4 {(2)}^{3} + 4 {(2)}^{2} + 15 (2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f (2) = - 4 \cdot 8 + 4 {(2)}^{2} + 15 (2)$

Step 4.2.1.2

Multiply $- 4$ by $8$ .

$f (2) = - 32 + 4 {(2)}^{2} + 15 (2)$

Step 4.2.1.3

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

$f (2) = - 32 + 4 \cdot 4 + 15 (2)$

Step 4.2.1.4

Multiply $4$ by $4$ .

$f (2) = - 32 + 16 + 15 (2)$

Step 4.2.1.5

Multiply $15$ by $2$ .

$f (2) = - 32 + 16 + 30$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $- 32$ and $16$ .

$f (2) = - 16 + 30$

Step 4.2.2.2

Add $- 16$ and $30$ .

$f (2) = 14$

Step 4.2.3

The final answer is $14$ .

$14$

Step 4.3

Convert $14$ to decimal.

$y = 14$

Step 5

Find the point at $x = 3$ .

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Step 5.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = - 4 {(3)}^{3} + 4 {(3)}^{2} + 15 (3)$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (3) = - 4 \cdot 27 + 4 {(3)}^{2} + 15 (3)$

Step 5.2.1.2

Multiply $- 4$ by $27$ .

$f (3) = - 108 + 4 {(3)}^{2} + 15 (3)$

Step 5.2.1.3

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

$f (3) = - 108 + 4 \cdot 9 + 15 (3)$

Step 5.2.1.4

Multiply $4$ by $9$ .

$f (3) = - 108 + 36 + 15 (3)$

Step 5.2.1.5

Multiply $15$ by $3$ .

$f (3) = - 108 + 36 + 45$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 108$ and $36$ .

$f (3) = - 72 + 45$

Step 5.2.2.2

Add $- 72$ and $45$ .

$f (3) = - 27$

Step 5.2.3

The final answer is $- 27$ .

$- 27$

Step 5.3

Convert $- 27$ to decimal.

$y = - 27$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 1 & - 7 \\ 0 & 0 \\ 1 & 15 \\ 2 & 14 \\ 3 & - 27 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 1 & - 7 \\ 0 & 0 \\ 1 & 15 \\ 2 & 14 \\ 3 & - 27 \end{array}$

Step 8