Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$f (x) = (x + 7) (x + 7) - 1$

Step 1.2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = x (x + 7) + 7 (x + 7) - 1$

Step 1.2.2

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 7 + 7 (x + 7) - 1$

Step 1.2.3

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 - 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $7$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $7$ by $7$ .

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

Step 1.3.2

Add $7 x$ and $7 x$ .

$f (x) = x^{2} + 14 x + 49 - 1$

Step 2

Subtract $1$ from $49$ .

$f (x) = x^{2} + 14 x + 48$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

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Step 4.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{14}{2 (1)}$

Step 4.2

Remove parentheses.

$x = - \frac{14}{2 (1)}$

Step 4.3

Simplify $- \frac{14}{2 (1)}$ .

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Step 4.3.1

Cancel the common factor of $14$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $14$ .

$x = - \frac{2 \cdot 7}{2 \cdot 1}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 \cdot 1$ .

$x = - \frac{2 \cdot 7}{2 (1)}$

Step 4.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 7}{2 \cdot 1}$

Step 4.3.1.2.3

Rewrite the expression.

$x = - \frac{7}{1}$

Step 4.3.1.2.4

Divide $7$ by $1$ .

$x = - 1 \cdot 7$

Step 4.3.2

Multiply $- 1$ by $7$ .

$x = - 7$

Step 5

Evaluate $f (- 7)$ .

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Step 5.1

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = {(- 7)}^{2} + 14 (- 7) + 48$

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

$f (- 7) = 49 + 14 (- 7) + 48$

Step 5.2.1.2

Multiply $14$ by $- 7$ .

$f (- 7) = 49 - 98 + 48$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $98$ from $49$ .

$f (- 7) = - 49 + 48$

Step 5.2.2.2

Add $- 49$ and $48$ .

$f (- 7) = - 1$

Step 5.2.3

The final answer is $- 1$ .

$- 1$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(- 7, - 1)$

Step 7