Algebra Examples

$y = - (x - 1) {(2 x + 5)}^{2}$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

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

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 $- 3$ .

$f (- 3) = 3 {(- 6 + 5)}^{2} + {(2 (- 3) + 5)}^{2}$

Step 1.2.1.2

Add $- 6$ and $5$ .

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Multiply $3$ by $1$ .

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

Step 1.2.1.5

Multiply $2$ by $- 3$ .

$f (- 3) = 3 + {(- 6 + 5)}^{2}$

Step 1.2.1.6

Add $- 6$ and $5$ .

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

Step 1.2.1.7

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

$f (- 3) = 3 + 1$

Step 1.2.2

Add $3$ and $1$ .

$f (- 3) = 4$

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.

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

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 $2$ by $- 2$ .

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

Step 2.2.1.2

Add $- 4$ and $5$ .

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

Step 2.2.1.3

One to any power is one.

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

Step 2.2.1.4

Multiply $2$ by $1$ .

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

Step 2.2.1.5

Multiply $2$ by $- 2$ .

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

Step 2.2.1.6

Add $- 4$ and $5$ .

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

Step 2.2.1.7

One to any power is one.

$f (- 2) = 2 + 1$

Step 2.2.2

Add $2$ and $1$ .

$f (- 2) = 3$

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 $x = - 1$ .

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Step 3.1

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

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

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 ${(2 (- 1) + 5)}^{2}$ by $1$ .

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

Step 3.2.1.2

Multiply $2$ by $- 1$ .

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

Step 3.2.1.3

Add $- 2$ and $5$ .

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

Step 3.2.1.4

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

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

Step 3.2.1.5

Multiply $2$ by $- 1$ .

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

Step 3.2.1.6

Add $- 2$ and $5$ .

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

Step 3.2.1.7

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

$f (- 1) = 9 + 9$

Step 3.2.2

Add $9$ and $9$ .

$f (- 1) = 18$

Step 3.2.3

The final answer is $18$ .

$18$

Step 3.3

Convert $18$ to decimal.

$y = 18$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = 0 {(2 (0) + 5)}^{2} + {(2 (0) + 5)}^{2}$

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 $0$ .

$f (0) = 0 {(0 + 5)}^{2} + {(2 (0) + 5)}^{2}$

Step 4.2.1.2

Add $0$ and $5$ .

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

Step 4.2.1.3

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

$f (0) = 0 \cdot 25 + {(2 (0) + 5)}^{2}$

Step 4.2.1.4

Multiply $0$ by $25$ .

$f (0) = 0 + {(2 (0) + 5)}^{2}$

Step 4.2.1.5

Multiply $2$ by $0$ .

$f (0) = 0 + {(0 + 5)}^{2}$

Step 4.2.1.6

Add $0$ and $5$ .

$f (0) = 0 + 5^{2}$

Step 4.2.1.7

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

$f (0) = 0 + 25$

Step 4.2.2

Add $0$ and $25$ .

$f (0) = 25$

Step 4.2.3

The final answer is $25$ .

$25$

Step 4.3

Convert $25$ to decimal.

$y = 25$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

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

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 $1$ .

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

Step 5.2.1.2

Add $2$ and $5$ .

$f (1) = - 1 \cdot 7^{2} + {(2 (1) + 5)}^{2}$

Step 5.2.1.3

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

$f (1) = - 1 \cdot 49 + {(2 (1) + 5)}^{2}$

Step 5.2.1.4

Multiply $- 1$ by $49$ .

$f (1) = - 49 + {(2 (1) + 5)}^{2}$

Step 5.2.1.5

Multiply $2$ by $1$ .

$f (1) = - 49 + {(2 + 5)}^{2}$

Step 5.2.1.6

Add $2$ and $5$ .

$f (1) = - 49 + 7^{2}$

Step 5.2.1.7

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

$f (1) = - 49 + 49$

Step 5.2.2

Add $- 49$ and $49$ .

$f (1) = 0$

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.

$\begin{array}{c} x & y \\ - 3 & 4 \\ - 2 & 3 \\ - 1 & 18 \\ 0 & 25 \\ 1 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 4 \\ - 2 & 3 \\ - 1 & 18 \\ 0 & 25 \\ 1 & 0 \end{array}$

Step 8