Calculus Examples

$y = x^{2} + 2 x - 3$

Step 1

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 2

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.

$x = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$x = - 1$

Step 3

Evaluate $f (- 1)$ .

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

$f (- 1) = {(- 1)}^{2} + 2 (- 1) - 3$

Simplify the result.

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Simplify each term.

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Raise $- 1$ to the power of $2$ .

$f (- 1) = 1 + 2 (- 1) - 3$

Multiply $2$ by $- 1$ .

$f (- 1) = 1 - 2 - 3$

Simplify by subtracting numbers.

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Subtract $2$ from $1$ .

$f (- 1) = - 1 - 3$

Subtract $3$ from $- 1$ .

$f (- 1) = - 4$

The final answer is $- 4$ .

$- 4$

Step 4

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

$(- 1, - 4)$

Step 5