Algebra Examples

$y = (2 x + 1) (2 x + 3) (2 x - 5)$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

$f (- 1) = 8 {(- 1)}^{3} - 4 {(- 1)}^{2} - 34 \cdot - 1 - 15$

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 $3$ .

$f (- 1) = 8 \cdot - 1 - 4 {(- 1)}^{2} - 34 \cdot - 1 - 15$

Step 1.2.1.2

Multiply $8$ by $- 1$ .

$f (- 1) = - 8 - 4 {(- 1)}^{2} - 34 \cdot - 1 - 15$

Step 1.2.1.3

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

$f (- 1) = - 8 - 4 \cdot 1 - 34 \cdot - 1 - 15$

Step 1.2.1.4

Multiply $- 4$ by $1$ .

$f (- 1) = - 8 - 4 - 34 \cdot - 1 - 15$

Step 1.2.1.5

Multiply $- 34$ by $- 1$ .

$f (- 1) = - 8 - 4 + 34 - 15$

Step 1.2.2

Simplify by adding and subtracting.

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Step 1.2.2.1

Subtract $4$ from $- 8$ .

$f (- 1) = - 12 + 34 - 15$

Step 1.2.2.2

Add $- 12$ and $34$ .

$f (- 1) = 22 - 15$

Step 1.2.2.3

Subtract $15$ from $22$ .

$f (- 1) = 7$

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.

$f (0) = 8 {(0)}^{3} - 4 {(0)}^{2} - 34 \cdot 0 - 15$

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 $0$ .

$f (0) = 8 \cdot 0 - 4 {(0)}^{2} - 34 \cdot 0 - 15$

Step 2.2.1.2

Multiply $8$ by $0$ .

$f (0) = 0 - 4 {(0)}^{2} - 34 \cdot 0 - 15$

Step 2.2.1.3

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

$f (0) = 0 - 4 \cdot 0 - 34 \cdot 0 - 15$

Step 2.2.1.4

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0 - 34 \cdot 0 - 15$

Step 2.2.1.5

Multiply $- 34$ by $0$ .

$f (0) = 0 + 0 + 0 - 15$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 15$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 - 15$

Step 2.2.2.3

Subtract $15$ from $0$ .

$f (0) = - 15$

Step 2.2.3

The final answer is $- 15$ .

$- 15$

Step 2.3

Convert $- 15$ to decimal.

$y = - 15$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = 8 {(1)}^{3} - 4 {(1)}^{2} - 34 \cdot 1 - 15$

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.

$f (1) = 8 \cdot 1 - 4 {(1)}^{2} - 34 \cdot 1 - 15$

Step 3.2.1.2

Multiply $8$ by $1$ .

$f (1) = 8 - 4 {(1)}^{2} - 34 \cdot 1 - 15$

Step 3.2.1.3

One to any power is one.

$f (1) = 8 - 4 \cdot 1 - 34 \cdot 1 - 15$

Step 3.2.1.4

Multiply $- 4$ by $1$ .

$f (1) = 8 - 4 - 34 \cdot 1 - 15$

Step 3.2.1.5

Multiply $- 34$ by $1$ .

$f (1) = 8 - 4 - 34 - 15$

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $4$ from $8$ .

$f (1) = 4 - 34 - 15$

Step 3.2.2.2

Subtract $34$ from $4$ .

$f (1) = - 30 - 15$

Step 3.2.2.3

Subtract $15$ from $- 30$ .

$f (1) = - 45$

Step 3.2.3

The final answer is $- 45$ .

$- 45$

Step 3.3

Convert $- 45$ to decimal.

$y = - 45$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = 8 {(2)}^{3} - 4 {(2)}^{2} - 34 \cdot 2 - 15$

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 $3$ .

$f (2) = 8 \cdot 8 - 4 {(2)}^{2} - 34 \cdot 2 - 15$

Step 4.2.1.2

Multiply $8$ by $8$ .

$f (2) = 64 - 4 {(2)}^{2} - 34 \cdot 2 - 15$

Step 4.2.1.3

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

$f (2) = 64 - 4 \cdot 4 - 34 \cdot 2 - 15$

Step 4.2.1.4

Multiply $- 4$ by $4$ .

$f (2) = 64 - 16 - 34 \cdot 2 - 15$

Step 4.2.1.5

Multiply $- 34$ by $2$ .

$f (2) = 64 - 16 - 68 - 15$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $16$ from $64$ .

$f (2) = 48 - 68 - 15$

Step 4.2.2.2

Subtract $68$ from $48$ .

$f (2) = - 20 - 15$

Step 4.2.2.3

Subtract $15$ from $- 20$ .

$f (2) = - 35$

Step 4.2.3

The final answer is $- 35$ .

$- 35$

Step 4.3

Convert $- 35$ to decimal.

$y = - 35$

Step 5

Find the point at $x = 3$ .

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Step 5.1

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

$f (3) = 8 {(3)}^{3} - 4 {(3)}^{2} - 34 \cdot 3 - 15$

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 $3$ .

$f (3) = 8 \cdot 27 - 4 {(3)}^{2} - 34 \cdot 3 - 15$

Step 5.2.1.2

Multiply $8$ by $27$ .

$f (3) = 216 - 4 {(3)}^{2} - 34 \cdot 3 - 15$

Step 5.2.1.3

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

$f (3) = 216 - 4 \cdot 9 - 34 \cdot 3 - 15$

Step 5.2.1.4

Multiply $- 4$ by $9$ .

$f (3) = 216 - 36 - 34 \cdot 3 - 15$

Step 5.2.1.5

Multiply $- 34$ by $3$ .

$f (3) = 216 - 36 - 102 - 15$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $36$ from $216$ .

$f (3) = 180 - 102 - 15$

Step 5.2.2.2

Subtract $102$ from $180$ .

$f (3) = 78 - 15$

Step 5.2.2.3

Subtract $15$ from $78$ .

$f (3) = 63$

Step 5.2.3

The final answer is $63$ .

$63$

Step 5.3

Convert $63$ to decimal.

$y = 63$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 7 \\ 0 & - 15 \\ 1 & - 45 \\ 2 & - 35 \\ 3 & 63 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 7 \\ 0 & - 15 \\ 1 & - 45 \\ 2 & - 35 \\ 3 & 63 \end{array}$

Step 8