Precalculus Examples

$P (x) = (x - 1) (x + 1) (x - 4)$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

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

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 $0$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0 - (0) + 4$

Step 1.2.1.4

Multiply $- 1$ by $0$ .

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

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $0$ and $0$ .

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

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 4$

Step 1.2.2.3

Add $0$ and $4$ .

$f (0) = 4$

Step 1.2.3

The final answer is $4$ .

$4$

Step 1.3

Convert $4$ to decimal.

$y = 4$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

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.

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

Step 2.2.1.2

One to any power is one.

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

Step 2.2.1.3

Multiply $- 4$ by $1$ .

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

Step 2.2.1.4

Multiply $- 1$ by $1$ .

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

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $4$ from $1$ .

$f (1) = - 3 - 1 + 4$

Step 2.2.2.2

Subtract $1$ from $- 3$ .

$f (1) = - 4 + 4$

Step 2.2.2.3

Add $- 4$ and $4$ .

$f (1) = 0$

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 $x = 2$ .

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Step 3.1

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

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

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 $3$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply $- 4$ by $4$ .

$f (2) = 8 - 16 - (2) + 4$

Step 3.2.1.4

Multiply $- 1$ by $2$ .

$f (2) = 8 - 16 - 2 + 4$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $16$ from $8$ .

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

Step 3.2.2.2

Subtract $2$ from $- 8$ .

$f (2) = - 10 + 4$

Step 3.2.2.3

Add $- 10$ and $4$ .

$f (2) = - 6$

Step 3.2.3

The final answer is $- 6$ .

$- 6$

Step 3.3

Convert $- 6$ to decimal.

$y = - 6$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

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 $3$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $- 4$ by $9$ .

$f (3) = 27 - 36 - (3) + 4$

Step 4.2.1.4

Multiply $- 1$ by $3$ .

$f (3) = 27 - 36 - 3 + 4$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $36$ from $27$ .

$f (3) = - 9 - 3 + 4$

Step 4.2.2.2

Subtract $3$ from $- 9$ .

$f (3) = - 12 + 4$

Step 4.2.2.3

Add $- 12$ and $4$ .

$f (3) = - 8$

Step 4.2.3

The final answer is $- 8$ .

$- 8$

Step 4.3

Convert $- 8$ to decimal.

$y = - 8$

Step 5

Find the point at $x = 4$ .

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Step 5.1

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

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

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 $3$ .

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

Step 5.2.1.2

Raise $4$ to the power of $2$ .

$f (4) = 64 - 4 \cdot 16 - (4) + 4$

Step 5.2.1.3

Multiply $- 4$ by $16$ .

$f (4) = 64 - 64 - (4) + 4$

Step 5.2.1.4

Multiply $- 1$ by $4$ .

$f (4) = 64 - 64 - 4 + 4$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $64$ from $64$ .

$f (4) = 0 - 4 + 4$

Step 5.2.2.2

Subtract $4$ from $0$ .

$f (4) = - 4 + 4$

Step 5.2.2.3

Add $- 4$ and $4$ .

$f (4) = 0$

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.

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 0 \\ 2 & - 6 \\ 3 & - 8 \\ 4 & 0 \end{array}$

Step 7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 0 \\ 2 & - 6 \\ 3 & - 8 \\ 4 & 0 \end{array}$

Step 8