Finite Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$f (x) = - ((x - 3) (x - 3)) + 2$

Step 1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = - (x (x - 3) - 3 (x - 3)) + 2$

Step 1.2.2

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 2$

Step 1.2.3

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 2$

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 $x$ .

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

Step 1.3.1.2

Move $- 3$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 3$ by $- 3$ .

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

Step 1.3.2

Subtract $3 x$ from $- 3 x$ .

$f (x) = - (x^{2} - 6 x + 9) + 2$

Step 1.4

Apply the distributive property.

$f (x) = - x^{2} - (- 6 x) - 1 \cdot 9 + 2$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 6$ by $- 1$ .

$f (x) = - x^{2} + 6 x - 1 \cdot 9 + 2$

Step 1.5.2

Multiply $- 1$ by $9$ .

$f (x) = - x^{2} + 6 x - 9 + 2$

Step 2

Add $- 9$ and $2$ .

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

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 $b$ .

$x = - \frac{6}{2 (- 1)}$

Step 4.2

Remove parentheses.

$x = - \frac{6}{2 (- 1)}$

Step 4.3

Simplify $- \frac{6}{2 (- 1)}$ .

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Step 4.3.1

Cancel the common factor of $6$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $6$ .

$x = - \frac{2 \cdot 3}{2 \cdot - 1}$

Step 4.3.1.2

Move the negative one from the denominator of $\frac{3}{- 1}$ .

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

Step 4.3.2

Multiply $- (- 1 \cdot 3)$ .

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Step 4.3.2.1

Multiply $- 1$ by $3$ .

$x = - - 3$

Step 4.3.2.2

Multiply $- 1$ by $- 3$ .

$x = 3$

Step 5

Evaluate $f (3)$ .

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Step 5.1

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

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

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 $3$ to the power of $2$ .

$f (3) = - 1 \cdot 9 + 6 (3) - 7$

Step 5.2.1.2

Multiply $- 1$ by $9$ .

$f (3) = - 9 + 6 (3) - 7$

Step 5.2.1.3

Multiply $6$ by $3$ .

$f (3) = - 9 + 18 - 7$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 9$ and $18$ .

$f (3) = 9 - 7$

Step 5.2.2.2

Subtract $7$ from $9$ .

$f (3) = 2$

Step 5.2.3

The final answer is $2$ .

$2$

Step 6

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

$(3, 2)$

Step 7