Algebra Examples

$y = (x - 1) {(2 x + 7)}^{2}$

Step 1

Find the point at $x = - 4$ .

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Step 1.1

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

$f (- 4) = (- 4) {(2 (- 4) + 7)}^{2} - {(2 (- 4) + 7)}^{2}$

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 $- 4$ .

$f (- 4) = - 4 {(- 8 + 7)}^{2} - {(2 (- 4) + 7)}^{2}$

Step 1.2.1.2

Add $- 8$ and $7$ .

$f (- 4) = - 4 {(- 1)}^{2} - {(2 (- 4) + 7)}^{2}$

Step 1.2.1.3

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

$f (- 4) = - 4 \cdot 1 - {(2 (- 4) + 7)}^{2}$

Step 1.2.1.4

Multiply $- 4$ by $1$ .

$f (- 4) = - 4 - {(2 (- 4) + 7)}^{2}$

Step 1.2.1.5

Multiply $2$ by $- 4$ .

$f (- 4) = - 4 - {(- 8 + 7)}^{2}$

Step 1.2.1.6

Add $- 8$ and $7$ .

$f (- 4) = - 4 - {(- 1)}^{2}$

Step 1.2.1.7

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.2.1.7.1

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

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Step 1.2.1.7.1.1

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

$f (- 4) = - 4 + (- 1) {(- 1)}^{2}$

Step 1.2.1.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 4) = - 4 + {(- 1)}^{1 + 2}$

Step 1.2.1.7.2

Add $1$ and $2$ .

$f (- 4) = - 4 + {(- 1)}^{3}$

Step 1.2.1.8

Raise $- 1$ to the power of $3$ .

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

Step 1.2.2

Subtract $1$ from $- 4$ .

$f (- 4) = - 5$

Step 1.2.3

The final answer is $- 5$ .

$- 5$

Step 1.3

Convert $- 5$ to decimal.

$y = - 5$

Step 2

Find the point at $x = - 3$ .

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Step 2.1

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

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

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 $- 3$ .

$f (- 3) = - 3 {(- 6 + 7)}^{2} - {(2 (- 3) + 7)}^{2}$

Step 2.2.1.2

Add $- 6$ and $7$ .

$f (- 3) = - 3 \cdot 1^{2} - {(2 (- 3) + 7)}^{2}$

Step 2.2.1.3

One to any power is one.

$f (- 3) = - 3 \cdot 1 - {(2 (- 3) + 7)}^{2}$

Step 2.2.1.4

Multiply $- 3$ by $1$ .

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

Step 2.2.1.5

Multiply $2$ by $- 3$ .

$f (- 3) = - 3 - {(- 6 + 7)}^{2}$

Step 2.2.1.6

Add $- 6$ and $7$ .

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

Step 2.2.1.7

One to any power is one.

$f (- 3) = - 3 - 1 \cdot 1$

Step 2.2.1.8

Multiply $- 1$ by $1$ .

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

Step 2.2.2

Subtract $1$ from $- 3$ .

$f (- 3) = - 4$

Step 2.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3

Convert $- 4$ to decimal.

$y = - 4$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

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

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

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 $- 2$ .

$f (- 2) = - 2 {(- 4 + 7)}^{2} - {(2 (- 2) + 7)}^{2}$

Step 3.2.1.2

Add $- 4$ and $7$ .

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

Step 3.2.1.3

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

$f (- 2) = - 2 \cdot 9 - {(2 (- 2) + 7)}^{2}$

Step 3.2.1.4

Multiply $- 2$ by $9$ .

$f (- 2) = - 18 - {(2 (- 2) + 7)}^{2}$

Step 3.2.1.5

Multiply $2$ by $- 2$ .

$f (- 2) = - 18 - {(- 4 + 7)}^{2}$

Step 3.2.1.6

Add $- 4$ and $7$ .

$f (- 2) = - 18 - 3^{2}$

Step 3.2.1.7

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

$f (- 2) = - 18 - 1 \cdot 9$

Step 3.2.1.8

Multiply $- 1$ by $9$ .

$f (- 2) = - 18 - 9$

Step 3.2.2

Subtract $9$ from $- 18$ .

$f (- 2) = - 27$

Step 3.2.3

The final answer is $- 27$ .

$- 27$

Step 3.3

Convert $- 27$ to decimal.

$y = - 27$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

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

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

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 $- 1$ .

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

Step 4.2.1.2

Add $- 2$ and $7$ .

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

Step 4.2.1.3

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

$f (- 1) = - 1 \cdot 25 - {(2 (- 1) + 7)}^{2}$

Step 4.2.1.4

Multiply $- 1$ by $25$ .

$f (- 1) = - 25 - {(2 (- 1) + 7)}^{2}$

Step 4.2.1.5

Multiply $2$ by $- 1$ .

$f (- 1) = - 25 - {(- 2 + 7)}^{2}$

Step 4.2.1.6

Add $- 2$ and $7$ .

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

Step 4.2.1.7

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

$f (- 1) = - 25 - 1 \cdot 25$

Step 4.2.1.8

Multiply $- 1$ by $25$ .

$f (- 1) = - 25 - 25$

Step 4.2.2

Subtract $25$ from $- 25$ .

$f (- 1) = - 50$

Step 4.2.3

The final answer is $- 50$ .

$- 50$

Step 4.3

Convert $- 50$ to decimal.

$y = - 50$

Step 5

Find the point at $x = 0$ .

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Step 5.1

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

$f (0) = (0) {(2 (0) + 7)}^{2} - {(2 (0) + 7)}^{2}$

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 $0$ .

$f (0) = 0 {(0 + 7)}^{2} - {(2 (0) + 7)}^{2}$

Step 5.2.1.2

Add $0$ and $7$ .

$f (0) = 0 \cdot 7^{2} - {(2 (0) + 7)}^{2}$

Step 5.2.1.3

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

$f (0) = 0 \cdot 49 - {(2 (0) + 7)}^{2}$

Step 5.2.1.4

Multiply $0$ by $49$ .

$f (0) = 0 - {(2 (0) + 7)}^{2}$

Step 5.2.1.5

Multiply $2$ by $0$ .

$f (0) = 0 - {(0 + 7)}^{2}$

Step 5.2.1.6

Add $0$ and $7$ .

$f (0) = 0 - 7^{2}$

Step 5.2.1.7

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

$f (0) = 0 - 1 \cdot 49$

Step 5.2.1.8

Multiply $- 1$ by $49$ .

$f (0) = 0 - 49$

Step 5.2.2

Subtract $49$ from $0$ .

$f (0) = - 49$

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.

$\begin{array}{c} x & y \\ - 4 & - 5 \\ - 3 & - 4 \\ - 2 & - 27 \\ - 1 & - 50 \\ 0 & - 49 \end{array}$

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

$\begin{array}{c} x & y \\ - 4 & - 5 \\ - 3 & - 4 \\ - 2 & - 27 \\ - 1 & - 50 \\ 0 & - 49 \end{array}$

Step 8