Algebra Examples

$y = - x^{2} + 2 x + 1$

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 $x = - \frac{b}{2 a}$

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

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

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

Remove parentheses.

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

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

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply.

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Multiply $- 1$ by $1$ .

$x = - - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Evaluate $f (1)$ .

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

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

Simplify the result.

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Simplify each term.

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One to any power is one.

$f (1) = - 1 \cdot 1 + 2 (1) + 1$

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 2 (1) + 1$

Multiply $2$ by $1$ .

$f (1) = - 1 + 2 + 1$

Simplify by adding numbers.

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Add $- 1$ and $2$ .

$f (1) = 1 + 1$

Add $1$ and $1$ .

$f (1) = 2$

The final answer is $2$ .

$2$

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

$(1, 2)$