Precalculus Examples

$y = x^{3} - 3 x^{2} - 6 x + 5$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = {(- 1)}^{3} - 3 {(- 1)}^{2} - 6 \cdot - 1 + 5$

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 $3$ .

$f (- 1) = - 1 - 3 {(- 1)}^{2} - 6 \cdot - 1 + 5$

Step 1.2.1.2

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

$f (- 1) = - 1 - 3 \cdot 1 - 6 \cdot - 1 + 5$

Step 1.2.1.3

Multiply $- 3$ by $1$ .

$f (- 1) = - 1 - 3 - 6 \cdot - 1 + 5$

Step 1.2.1.4

Multiply $- 6$ by $- 1$ .

$f (- 1) = - 1 - 3 + 6 + 5$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $3$ from $- 1$ .

$f (- 1) = - 4 + 6 + 5$

Step 1.2.2.2

Add $- 4$ and $6$ .

$f (- 1) = 2 + 5$

Step 1.2.2.3

Add $2$ and $5$ .

$f (- 1) = 7$

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.

$f (0) = {(0)}^{3} - 3 {(0)}^{2} - 6 \cdot 0 + 5$

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 $0$ .

$f (0) = 0 - 3 {(0)}^{2} - 6 \cdot 0 + 5$

Step 2.2.1.2

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

$f (0) = 0 - 3 \cdot 0 - 6 \cdot 0 + 5$

Step 2.2.1.3

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 6 \cdot 0 + 5$

Step 2.2.1.4

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 + 0 + 5$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 5$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 5$

Step 2.2.2.3

Add $0$ and $5$ .

$f (0) = 5$

Step 2.2.3

The final answer is $5$ .

$5$

Step 2.3

Convert $5$ to decimal.

$y = 5$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = {(1)}^{3} - 3 {(1)}^{2} - 6 \cdot 1 + 5$

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.

$f (1) = 1 - 3 {(1)}^{2} - 6 \cdot 1 + 5$

Step 3.2.1.2

One to any power is one.

$f (1) = 1 - 3 \cdot 1 - 6 \cdot 1 + 5$

Step 3.2.1.3

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 - 6 \cdot 1 + 5$

Step 3.2.1.4

Multiply $- 6$ by $1$ .

$f (1) = 1 - 3 - 6 + 5$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $3$ from $1$ .

$f (1) = - 2 - 6 + 5$

Step 3.2.2.2

Subtract $6$ from $- 2$ .

$f (1) = - 8 + 5$

Step 3.2.2.3

Add $- 8$ and $5$ .

$f (1) = - 3$

Step 3.2.3

The final answer is $- 3$ .

$- 3$

Step 3.3

Convert $- 3$ to decimal.

$y = - 3$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = {(2)}^{3} - 3 {(2)}^{2} - 6 \cdot 2 + 5$

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 $3$ .

$f (2) = 8 - 3 {(2)}^{2} - 6 \cdot 2 + 5$

Step 4.2.1.2

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

$f (2) = 8 - 3 \cdot 4 - 6 \cdot 2 + 5$

Step 4.2.1.3

Multiply $- 3$ by $4$ .

$f (2) = 8 - 12 - 6 \cdot 2 + 5$

Step 4.2.1.4

Multiply $- 6$ by $2$ .

$f (2) = 8 - 12 - 12 + 5$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $12$ from $8$ .

$f (2) = - 4 - 12 + 5$

Step 4.2.2.2

Subtract $12$ from $- 4$ .

$f (2) = - 16 + 5$

Step 4.2.2.3

Add $- 16$ and $5$ .

$f (2) = - 11$

Step 4.2.3

The final answer is $- 11$ .

$- 11$

Step 4.3

Convert $- 11$ to decimal.

$y = - 11$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = {(3)}^{3} - 3 {(3)}^{2} - 6 \cdot 3 + 5$

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 $3$ .

$f (3) = 27 - 3 {(3)}^{2} - 6 \cdot 3 + 5$

Step 5.2.1.2

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

$f (3) = 27 - 3 \cdot 9 - 6 \cdot 3 + 5$

Step 5.2.1.3

Multiply $- 3$ by $9$ .

$f (3) = 27 - 27 - 6 \cdot 3 + 5$

Step 5.2.1.4

Multiply $- 6$ by $3$ .

$f (3) = 27 - 27 - 18 + 5$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $27$ from $27$ .

$f (3) = 0 - 18 + 5$

Step 5.2.2.2

Subtract $18$ from $0$ .

$f (3) = - 18 + 5$

Step 5.2.2.3

Add $- 18$ and $5$ .

$f (3) = - 13$

Step 5.2.3

The final answer is $- 13$ .

$- 13$

Step 5.3

Convert $- 13$ to decimal.

$y = - 13$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 7 \\ 0 & 5 \\ 1 & - 3 \\ 2 & - 11 \\ 3 & - 13 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 7 \\ 0 & 5 \\ 1 & - 3 \\ 2 & - 11 \\ 3 & - 13 \end{array}$

Step 8