Find the Maximum/Minimum Value y=-x^2+4x-1

$y = - x^{2} + 4 x - 1$

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

Step 2

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

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

$x = - \frac{4}{2 (- 1)}$

Remove parentheses.

$x = - \frac{4}{2 (- 1)}$

Simplify $- \frac{4}{2 (- 1)}$ .

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$x = - - 2$

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 3

Evaluate $f (2)$ .

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

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

Simplify the result.

Simplify each term.

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

$f (2) = - 1 \cdot 4 + 4 (2) - 1$

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 4 (2) - 1$

Multiply $4$ by $2$ .

$f (2) = - 4 + 8 - 1$

Simplify by adding and subtracting.

Add $- 4$ and $8$ .

$f (2) = 4 - 1$

Subtract $1$ from $4$ .

$f (2) = 3$

The final answer is $3$ .

$3$

Step 4

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

$(2, 3)$

Step 5

Find the Maximum/Minimum Value y=-x^2+4x-1

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