Linear Algebra Examples

$y = x^{3} + 3 x^{2} - 9 x$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

$f (- 3) = {(- 3)}^{3} + 3 {(- 3)}^{2} - 9 \cdot - 3$

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

$f (- 3) = - 27 + 3 {(- 3)}^{2} - 9 \cdot - 3$

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $3$ by $9$ .

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

Step 1.2.1.4

Multiply $- 9$ by $- 3$ .

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

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 27$ and $27$ .

$f (- 3) = 0 + 27$

Step 1.2.2.2

Add $0$ and $27$ .

$f (- 3) = 27$

Step 1.2.3

The final answer is $27$ .

$27$

Step 1.3

Convert $27$ to decimal.

$y = 27$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = {(- 2)}^{3} + 3 {(- 2)}^{2} - 9 \cdot - 2$

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

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

$f (- 2) = - 8 + 3 {(- 2)}^{2} - 9 \cdot - 2$

Step 2.2.1.2

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

$f (- 2) = - 8 + 3 \cdot 4 - 9 \cdot - 2$

Step 2.2.1.3

Multiply $3$ by $4$ .

$f (- 2) = - 8 + 12 - 9 \cdot - 2$

Step 2.2.1.4

Multiply $- 9$ by $- 2$ .

$f (- 2) = - 8 + 12 + 18$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 8$ and $12$ .

$f (- 2) = 4 + 18$

Step 2.2.2.2

Add $4$ and $18$ .

$f (- 2) = 22$

Step 2.2.3

The final answer is $22$ .

$22$

Step 2.3

Convert $22$ to decimal.

$y = 22$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = {(- 1)}^{3} + 3 {(- 1)}^{2} - 9 \cdot - 1$

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

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

$f (- 1) = - 1 + 3 {(- 1)}^{2} - 9 \cdot - 1$

Step 3.2.1.2

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

$f (- 1) = - 1 + 3 \cdot 1 - 9 \cdot - 1$

Step 3.2.1.3

Multiply $3$ by $1$ .

$f (- 1) = - 1 + 3 - 9 \cdot - 1$

Step 3.2.1.4

Multiply $- 9$ by $- 1$ .

$f (- 1) = - 1 + 3 + 9$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $- 1$ and $3$ .

$f (- 1) = 2 + 9$

Step 3.2.2.2

Add $2$ and $9$ .

$f (- 1) = 11$

Step 3.2.3

The final answer is $11$ .

$11$

Step 3.3

Convert $11$ to decimal.

$y = 11$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = {(0)}^{3} + 3 {(0)}^{2} - 9 \cdot 0$

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

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

$f (0) = 0 + 3 {(0)}^{2} - 9 \cdot 0$

Step 4.2.1.2

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

$f (0) = 0 + 3 \cdot 0 - 9 \cdot 0$

Step 4.2.1.3

Multiply $3$ by $0$ .

$f (0) = 0 + 0 - 9 \cdot 0$

Step 4.2.1.4

Multiply $- 9$ by $0$ .

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

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.2.3

The final answer is $0$ .

$0$

Step 4.3

Convert $0$ to decimal.

$y = 0$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

$f (1) = {(1)}^{3} + 3 {(1)}^{2} - 9 \cdot 1$

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

One to any power is one.

$f (1) = 1 + 3 {(1)}^{2} - 9 \cdot 1$

Step 5.2.1.2

One to any power is one.

$f (1) = 1 + 3 \cdot 1 - 9 \cdot 1$

Step 5.2.1.3

Multiply $3$ by $1$ .

$f (1) = 1 + 3 - 9 \cdot 1$

Step 5.2.1.4

Multiply $- 9$ by $1$ .

$f (1) = 1 + 3 - 9$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $1$ and $3$ .

$f (1) = 4 - 9$

Step 5.2.2.2

Subtract $9$ from $4$ .

$f (1) = - 5$

Step 5.2.3

The final answer is $- 5$ .

$- 5$

Step 5.3

Convert $- 5$ to decimal.

$y = - 5$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 27 \\ - 2 & 22 \\ - 1 & 11 \\ 0 & 0 \\ 1 & - 5 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 27 \\ - 2 & 22 \\ - 1 & 11 \\ 0 & 0 \\ 1 & - 5 \end{array}$

Step 8