Algebra Examples

$y = - {(2 x + 3)}^{2} (x - 1)$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

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

Multiply $2$ by $- 2$ .

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

Step 1.2.1.2

Add $- 4$ and $3$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $2$ by $1$ .

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

Step 1.2.1.5

Multiply $2$ by $- 2$ .

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

Step 1.2.1.6

Add $- 4$ and $3$ .

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

Step 1.2.1.7

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

$f (- 2) = 2 + 1$

Step 1.2.2

Add $2$ and $1$ .

$f (- 2) = 3$

Step 1.2.3

The final answer is $3$ .

$3$

Step 1.3

Convert $3$ to decimal.

$y = 3$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

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

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

Multiply ${(2 (- 1) + 3)}^{2}$ by $1$ .

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

Step 2.2.1.2

Multiply $2$ by $- 1$ .

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

Step 2.2.1.3

Add $- 2$ and $3$ .

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

Step 2.2.1.4

One to any power is one.

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

Step 2.2.1.5

Multiply $2$ by $- 1$ .

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

Step 2.2.1.6

Add $- 2$ and $3$ .

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

Step 2.2.1.7

One to any power is one.

$f (- 1) = 1 + 1$

Step 2.2.2

Add $1$ and $1$ .

$f (- 1) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 0 {(2 (0) + 3)}^{2} + {(2 (0) + 3)}^{2}$

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

Multiply $2$ by $0$ .

$f (0) = 0 {(0 + 3)}^{2} + {(2 (0) + 3)}^{2}$

Step 3.2.1.2

Add $0$ and $3$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Multiply $0$ by $9$ .

$f (0) = 0 + {(2 (0) + 3)}^{2}$

Step 3.2.1.5

Multiply $2$ by $0$ .

$f (0) = 0 + {(0 + 3)}^{2}$

Step 3.2.1.6

Add $0$ and $3$ .

$f (0) = 0 + 3^{2}$

Step 3.2.1.7

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

$f (0) = 0 + 9$

Step 3.2.2

Add $0$ and $9$ .

$f (0) = 9$

Step 3.2.3

The final answer is $9$ .

$9$

Step 3.3

Convert $9$ to decimal.

$y = 9$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

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

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

Multiply $2$ by $1$ .

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

Step 4.2.1.2

Add $2$ and $3$ .

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

Step 4.2.1.3

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

$f (1) = - 1 \cdot 25 + {(2 (1) + 3)}^{2}$

Step 4.2.1.4

Multiply $- 1$ by $25$ .

$f (1) = - 25 + {(2 (1) + 3)}^{2}$

Step 4.2.1.5

Multiply $2$ by $1$ .

$f (1) = - 25 + {(2 + 3)}^{2}$

Step 4.2.1.6

Add $2$ and $3$ .

$f (1) = - 25 + 5^{2}$

Step 4.2.1.7

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

$f (1) = - 25 + 25$

Step 4.2.2

Add $- 25$ and $25$ .

$f (1) = 0$

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 $x = 2$ .

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Step 5.1

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

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

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

Multiply $2$ by $2$ .

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

Step 5.2.1.2

Add $4$ and $3$ .

$f (2) = - 2 \cdot 7^{2} + {(2 (2) + 3)}^{2}$

Step 5.2.1.3

Raise $7$ to the power of $2$ .

$f (2) = - 2 \cdot 49 + {(2 (2) + 3)}^{2}$

Step 5.2.1.4

Multiply $- 2$ by $49$ .

$f (2) = - 98 + {(2 (2) + 3)}^{2}$

Step 5.2.1.5

Multiply $2$ by $2$ .

$f (2) = - 98 + {(4 + 3)}^{2}$

Step 5.2.1.6

Add $4$ and $3$ .

$f (2) = - 98 + 7^{2}$

Step 5.2.1.7

Raise $7$ to the power of $2$ .

$f (2) = - 98 + 49$

Step 5.2.2

Add $- 98$ and $49$ .

$f (2) = - 49$

Step 5.2.3

The final answer is $- 49$ .

$- 49$

Step 5.3

Convert $- 49$ to decimal.

$y = - 49$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 2 \\ 0 & 9 \\ 1 & 0 \\ 2 & - 49 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 2 \\ 0 & 9 \\ 1 & 0 \\ 2 & - 49 \end{array}$

Step 8