Pre-Algebra Examples

$x^{3} - 8 x^{2} + 16 x - 5$

Step 1

Find the point at $x = 1$ .

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Step 1.1

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

$f (1) = {(1)}^{3} - 8 {(1)}^{2} + 16 (1) - 5$

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

One to any power is one.

$f (1) = 1 - 8 {(1)}^{2} + 16 (1) - 5$

Step 1.2.1.2

One to any power is one.

$f (1) = 1 - 8 \cdot 1 + 16 (1) - 5$

Step 1.2.1.3

Multiply $- 8$ by $1$ .

$f (1) = 1 - 8 + 16 (1) - 5$

Step 1.2.1.4

Multiply $16$ by $1$ .

$f (1) = 1 - 8 + 16 - 5$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $8$ from $1$ .

$f (1) = - 7 + 16 - 5$

Step 1.2.2.2

Add $- 7$ and $16$ .

$f (1) = 9 - 5$

Step 1.2.2.3

Subtract $5$ from $9$ .

$f (1) = 4$

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 $x = 2$ .

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Step 2.1

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

$f (2) = {(2)}^{3} - 8 {(2)}^{2} + 16 (2) - 5$

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

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

$f (2) = 8 - 8 {(2)}^{2} + 16 (2) - 5$

Step 2.2.1.2

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

$f (2) = 8 - 8 \cdot 4 + 16 (2) - 5$

Step 2.2.1.3

Multiply $- 8$ by $4$ .

$f (2) = 8 - 32 + 16 (2) - 5$

Step 2.2.1.4

Multiply $16$ by $2$ .

$f (2) = 8 - 32 + 32 - 5$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $32$ from $8$ .

$f (2) = - 24 + 32 - 5$

Step 2.2.2.2

Add $- 24$ and $32$ .

$f (2) = 8 - 5$

Step 2.2.2.3

Subtract $5$ from $8$ .

$f (2) = 3$

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 $x = 3$ .

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Step 3.1

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

$f (3) = {(3)}^{3} - 8 {(3)}^{2} + 16 (3) - 5$

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

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

$f (3) = 27 - 8 {(3)}^{2} + 16 (3) - 5$

Step 3.2.1.2

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

$f (3) = 27 - 8 \cdot 9 + 16 (3) - 5$

Step 3.2.1.3

Multiply $- 8$ by $9$ .

$f (3) = 27 - 72 + 16 (3) - 5$

Step 3.2.1.4

Multiply $16$ by $3$ .

$f (3) = 27 - 72 + 48 - 5$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $72$ from $27$ .

$f (3) = - 45 + 48 - 5$

Step 3.2.2.2

Add $- 45$ and $48$ .

$f (3) = 3 - 5$

Step 3.2.2.3

Subtract $5$ from $3$ .

$f (3) = - 2$

Step 3.2.3

The final answer is $- 2$ .

$- 2$

Step 3.3

Convert $- 2$ to decimal.

$y = - 2$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

$f (4) = {(4)}^{3} - 8 {(4)}^{2} + 16 (4) - 5$

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 $4$ to the power of $3$ .

$f (4) = 64 - 8 {(4)}^{2} + 16 (4) - 5$

Step 4.2.1.2

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

$f (4) = 64 - 8 \cdot 16 + 16 (4) - 5$

Step 4.2.1.3

Multiply $- 8$ by $16$ .

$f (4) = 64 - 128 + 16 (4) - 5$

Step 4.2.1.4

Multiply $16$ by $4$ .

$f (4) = 64 - 128 + 64 - 5$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $128$ from $64$ .

$f (4) = - 64 + 64 - 5$

Step 4.2.2.2

Add $- 64$ and $64$ .

$f (4) = 0 - 5$

Step 4.2.2.3

Subtract $5$ from $0$ .

$f (4) = - 5$

Step 4.2.3

The final answer is $- 5$ .

$- 5$

Step 4.3

Convert $- 5$ to decimal.

$y = - 5$

Step 5

Find the point at $x = 5$ .

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Step 5.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{3} - 8 {(5)}^{2} + 16 (5) - 5$

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 $5$ to the power of $3$ .

$f (5) = 125 - 8 {(5)}^{2} + 16 (5) - 5$

Step 5.2.1.2

Raise $5$ to the power of $2$ .

$f (5) = 125 - 8 \cdot 25 + 16 (5) - 5$

Step 5.2.1.3

Multiply $- 8$ by $25$ .

$f (5) = 125 - 200 + 16 (5) - 5$

Step 5.2.1.4

Multiply $16$ by $5$ .

$f (5) = 125 - 200 + 80 - 5$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $200$ from $125$ .

$f (5) = - 75 + 80 - 5$

Step 5.2.2.2

Add $- 75$ and $80$ .

$f (5) = 5 - 5$

Step 5.2.2.3

Subtract $5$ from $5$ .

$f (5) = 0$

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.

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

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

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

Step 8