Precalculus Examples

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

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

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

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

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

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

Step 1.2.1.2

Multiply $- 1$ by $0$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $4$ by $0$ .

$f (0) = 0 + 0$

Step 1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

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

One to any power is one.

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

Step 2.2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.2.1.3

One to any power is one.

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

Step 2.2.1.4

Multiply $4$ by $1$ .

$f (1) = - 1 + 4$

Step 2.2.2

Add $- 1$ and $4$ .

$f (1) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 2.3

Convert $3$ to decimal.

$y = 3$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

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

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

$f (2) = - 1 \cdot 8 + 4 {(2)}^{2}$

Step 3.2.1.2

Multiply $- 1$ by $8$ .

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

Step 3.2.1.3

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

$f (2) = - 8 + 4 \cdot 4$

Step 3.2.1.4

Multiply $4$ by $4$ .

$f (2) = - 8 + 16$

Step 3.2.2

Add $- 8$ and $16$ .

$f (2) = 8$

Step 3.2.3

The final answer is $8$ .

$8$

Step 3.3

Convert $8$ to decimal.

$y = 8$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

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 $3$ to the power of $3$ .

$f (3) = - 1 \cdot 27 + 4 {(3)}^{2}$

Step 4.2.1.2

Multiply $- 1$ by $27$ .

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

Step 4.2.1.3

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

$f (3) = - 27 + 4 \cdot 9$

Step 4.2.1.4

Multiply $4$ by $9$ .

$f (3) = - 27 + 36$

Step 4.2.2

Add $- 27$ and $36$ .

$f (3) = 9$

Step 4.2.3

The final answer is $9$ .

$9$

Step 4.3

Convert $9$ to decimal.

$y = 9$

Step 5

Find the point at $x = 4$ .

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Step 5.1

Replace the variable $x$ with $4$ in the expression.

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

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 $4$ to the power of $3$ .

$f (4) = - 1 \cdot 64 + 4 {(4)}^{2}$

Step 5.2.1.2

Multiply $- 1$ by $64$ .

$f (4) = - 64 + 4 {(4)}^{2}$

Step 5.2.1.3

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

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Step 5.2.1.3.1

Multiply $4$ by ${(4)}^{2}$ .

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Step 5.2.1.3.1.1

Raise $4$ to the power of $1$ .

$f (4) = - 64 + 4 {(4)}^{2}$

Step 5.2.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (4) = - 64 + 4^{1 + 2}$

Step 5.2.1.3.2

Add $1$ and $2$ .

$f (4) = - 64 + 4^{3}$

Step 5.2.1.4

Raise $4$ to the power of $3$ .

$f (4) = - 64 + 64$

Step 5.2.2

Add $- 64$ and $64$ .

$f (4) = 0$

Step 5.2.3

The final answer is $0$ .

$0$

Step 5.3

Convert $0$ to decimal.

$y = 0$

Step 6

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 3 \\ 2 & 8 \\ 3 & 9 \\ 4 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 3 \\ 2 & 8 \\ 3 & 9 \\ 4 & 0 \end{array}$

Step 8