Pre-Algebra Examples

$H (x) = - x^{3} + 5 x^{2} - 3 x - 9$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = - {(0)}^{3} + 5 {(0)}^{2} - 3 \cdot 0 - 9$

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 $0$ .

$f (0) = - 0 + 5 {(0)}^{2} - 3 \cdot 0 - 9$

Step 1.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 + 5 {(0)}^{2} - 3 \cdot 0 - 9$

Step 1.2.1.3

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

$f (0) = 0 + 5 \cdot 0 - 3 \cdot 0 - 9$

Step 1.2.1.4

Multiply $5$ by $0$ .

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

Step 1.2.1.5

Multiply $- 3$ by $0$ .

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

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Add $0$ and $0$ .

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

Step 1.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 9$

Step 1.2.2.3

Subtract $9$ from $0$ .

$f (0) = - 9$

Step 1.2.3

The final answer is $- 9$ .

$- 9$

Step 1.3

Convert $- 9$ to decimal.

$y = - 9$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = - {(1)}^{3} + 5 {(1)}^{2} - 3 \cdot 1 - 9$

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.

$f (1) = - 1 \cdot 1 + 5 {(1)}^{2} - 3 \cdot 1 - 9$

Step 2.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 5 {(1)}^{2} - 3 \cdot 1 - 9$

Step 2.2.1.3

One to any power is one.

$f (1) = - 1 + 5 \cdot 1 - 3 \cdot 1 - 9$

Step 2.2.1.4

Multiply $5$ by $1$ .

$f (1) = - 1 + 5 - 3 \cdot 1 - 9$

Step 2.2.1.5

Multiply $- 3$ by $1$ .

$f (1) = - 1 + 5 - 3 - 9$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 1$ and $5$ .

$f (1) = 4 - 3 - 9$

Step 2.2.2.2

Subtract $3$ from $4$ .

$f (1) = 1 - 9$

Step 2.2.2.3

Subtract $9$ from $1$ .

$f (1) = - 8$

Step 2.2.3

The final answer is $- 8$ .

$- 8$

Step 2.3

Convert $- 8$ to decimal.

$y = - 8$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = - {(2)}^{3} + 5 {(2)}^{2} - 3 \cdot 2 - 9$

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 $3$ .

$f (2) = - 1 \cdot 8 + 5 {(2)}^{2} - 3 \cdot 2 - 9$

Step 3.2.1.2

Multiply $- 1$ by $8$ .

$f (2) = - 8 + 5 {(2)}^{2} - 3 \cdot 2 - 9$

Step 3.2.1.3

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

$f (2) = - 8 + 5 \cdot 4 - 3 \cdot 2 - 9$

Step 3.2.1.4

Multiply $5$ by $4$ .

$f (2) = - 8 + 20 - 3 \cdot 2 - 9$

Step 3.2.1.5

Multiply $- 3$ by $2$ .

$f (2) = - 8 + 20 - 6 - 9$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 8$ and $20$ .

$f (2) = 12 - 6 - 9$

Step 3.2.2.2

Subtract $6$ from $12$ .

$f (2) = 6 - 9$

Step 3.2.2.3

Subtract $9$ from $6$ .

$f (2) = - 3$

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 $x = 3$ .

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Step 4.1

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

$f (3) = - {(3)}^{3} + 5 {(3)}^{2} - 3 \cdot 3 - 9$

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 $3$ .

$f (3) = - 1 \cdot 27 + 5 {(3)}^{2} - 3 \cdot 3 - 9$

Step 4.2.1.2

Multiply $- 1$ by $27$ .

$f (3) = - 27 + 5 {(3)}^{2} - 3 \cdot 3 - 9$

Step 4.2.1.3

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

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

Step 4.2.1.4

Multiply $5$ by $9$ .

$f (3) = - 27 + 45 - 3 \cdot 3 - 9$

Step 4.2.1.5

Multiply $- 3$ by $3$ .

$f (3) = - 27 + 45 - 9 - 9$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 27$ and $45$ .

$f (3) = 18 - 9 - 9$

Step 4.2.2.2

Subtract $9$ from $18$ .

$f (3) = 9 - 9$

Step 4.2.2.3

Subtract $9$ from $9$ .

$f (3) = 0$

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 $x = 4$ .

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Step 5.1

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

$f (4) = - {(4)}^{3} + 5 {(4)}^{2} - 3 \cdot 4 - 9$

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 $3$ .

$f (4) = - 1 \cdot 64 + 5 {(4)}^{2} - 3 \cdot 4 - 9$

Step 5.2.1.2

Multiply $- 1$ by $64$ .

$f (4) = - 64 + 5 {(4)}^{2} - 3 \cdot 4 - 9$

Step 5.2.1.3

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

$f (4) = - 64 + 5 \cdot 16 - 3 \cdot 4 - 9$

Step 5.2.1.4

Multiply $5$ by $16$ .

$f (4) = - 64 + 80 - 3 \cdot 4 - 9$

Step 5.2.1.5

Multiply $- 3$ by $4$ .

$f (4) = - 64 + 80 - 12 - 9$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 64$ and $80$ .

$f (4) = 16 - 12 - 9$

Step 5.2.2.2

Subtract $12$ from $16$ .

$f (4) = 4 - 9$

Step 5.2.2.3

Subtract $9$ from $4$ .

$f (4) = - 5$

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.

$\begin{array}{c} x & y \\ 0 & - 9 \\ 1 & - 8 \\ 2 & - 3 \\ 3 & 0 \\ 4 & - 5 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & - 9 \\ 1 & - 8 \\ 2 & - 3 \\ 3 & 0 \\ 4 & - 5 \end{array}$

Step 8