Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$f (x) = - ((x - 5) (x - 5)) + 3$

Step 1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = - (x (x - 5) - 5 (x - 5)) + 3$

Step 1.2.2

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 5 - 5 (x - 5)) + 3$

Step 1.2.3

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + 3$

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 $x$ .

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

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 5$ by $- 5$ .

$f (x) = - (x^{2} - 5 x - 5 x + 25) + 3$

Step 1.3.2

Subtract $5 x$ from $- 5 x$ .

$f (x) = - (x^{2} - 10 x + 25) + 3$

Step 1.4

Apply the distributive property.

$f (x) = - x^{2} - (- 10 x) - 1 \cdot 25 + 3$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 10$ by $- 1$ .

$f (x) = - x^{2} + 10 x - 1 \cdot 25 + 3$

Step 1.5.2

Multiply $- 1$ by $25$ .

$f (x) = - x^{2} + 10 x - 25 + 3$

Step 2

Add $- 25$ and $3$ .

$f (x) = - x^{2} + 10 x - 22$

Step 3

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

$f_{max}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{10}{2 (- 1)}$

Step 4.2

Remove parentheses.

$x = - \frac{10}{2 (- 1)}$

Step 4.3

Simplify $- \frac{10}{2 (- 1)}$ .

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Step 4.3.1

Cancel the common factor of $10$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $10$ .

$x = - \frac{2 \cdot 5}{2 \cdot - 1}$

Step 4.3.1.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$x = - (- 1 \cdot 5)$

Step 4.3.2

Multiply $- (- 1 \cdot 5)$ .

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Step 4.3.2.1

Multiply $- 1$ by $5$ .

$x = - - 5$

Step 4.3.2.2

Multiply $- 1$ by $- 5$ .

$x = 5$

Step 5

Evaluate $f (5)$ .

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Step 5.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - {(5)}^{2} + 10 (5) - 22$

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 $5$ to the power of $2$ .

$f (5) = - 1 \cdot 25 + 10 (5) - 22$

Step 5.2.1.2

Multiply $- 1$ by $25$ .

$f (5) = - 25 + 10 (5) - 22$

Step 5.2.1.3

Multiply $10$ by $5$ .

$f (5) = - 25 + 50 - 22$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 25$ and $50$ .

$f (5) = 25 - 22$

Step 5.2.2.2

Subtract $22$ from $25$ .

$f (5) = 3$

Step 5.2.3

The final answer is $3$ .

$3$

Step 6

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

$(5, 3)$

Step 7