Calculus Examples

$f (x) = a {(x - h)}^{2} + k$

Step 1

Simplify each term.

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Step 1.1

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

$f (x) = a ((x - h) (x - h)) + k$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f (x) = a (x (x - h) - h (x - h)) + k$

Step 1.2.2

Apply the distributive property.

$f (x) = a (x \cdot x + x (- h) - h (x - h)) + k$

Step 1.2.3

Apply the distributive property.

$f (x) = a (x \cdot x + x (- h) - h x - h (- h)) + k$

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 $x$ .

$f (x) = a (x^{2} + x (- h) - h x - h (- h)) + k$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$f (x) = a (x^{2} - x h - h x - h (- h)) + k$

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

$f (x) = a (x^{2} - x h - h x - 1 \cdot (- 1 h \cdot h)) + k$

Step 1.3.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 1.3.1.4.1

Move $h$ .

$f (x) = a (x^{2} - x h - h x - 1 \cdot (- 1 (h \cdot h))) + k$

Step 1.3.1.4.2

Multiply $h$ by $h$ .

$f (x) = a (x^{2} - x h - h x - 1 \cdot (- 1 h^{2})) + k$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$f (x) = a (x^{2} - x h - h x + 1 h^{2}) + k$

Step 1.3.1.6

Multiply $h^{2}$ by $1$ .

$f (x) = a (x^{2} - x h - h x + h^{2}) + k$

Step 1.3.2

Subtract $h x$ from $- x h$ .

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Step 1.3.2.1

Move $x$ .

$f (x) = a (x^{2} - h x - h x + h^{2}) + k$

Step 1.3.2.2

Subtract $h x$ from $- h x$ .

$f (x) = a (x^{2} - 2 h x + h^{2}) + k$

Step 1.4

Apply the distributive property.

$f (x) = a x^{2} + a (- 2 h x) + a h^{2} + k$

Step 1.5

Rewrite using the commutative property of multiplication.

$f (x) = a x^{2} - 2 a h x + a h^{2} + k$

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{0}{2 (1)}$

Step 3.2

Remove parentheses.

$x = - \frac{0}{2 (1)}$

Step 3.3

Simplify $- \frac{0}{2 (1)}$ .

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Step 3.3.1

Cancel the common factor of $0$ and $2$ .

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Step 3.3.1.1

Factor $2$ out of $0$ .

$x = - \frac{2 (0)}{2 (1)}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Cancel the common factor.

$x = - \frac{2 \cdot 0}{2 \cdot 1}$

Step 3.3.1.2.2

Rewrite the expression.

$x = - \frac{0}{1}$

Step 3.3.1.2.3

Divide $0$ by $1$ .

$x = - 0$

Step 3.3.2

Multiply $- 1$ by $0$ .

$x = 0$

Step 4

Evaluate $f (0)$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = a {(0)}^{2} - 2 a h \cdot 0 + a h^{2} + k$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = a \cdot 0 - 2 a h \cdot 0 + a h^{2} + k$

Step 4.2.1.2

Multiply $a$ by $0$ .

$f (0) = 0 - 2 a h \cdot 0 + a h^{2} + k$

Step 4.2.1.3

Multiply $- 2 a h \cdot 0$ .

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Step 4.2.1.3.1

Multiply $0$ by $- 2$ .

$f (0) = 0 + 0 a h + a h^{2} + k$

Step 4.2.1.3.2

Multiply $0$ by $a$ .

$f (0) = 0 + 0 h + a h^{2} + k$

Step 4.2.1.3.3

Multiply $0$ by $h$ .

$f (0) = 0 + 0 + a h^{2} + k$

Step 4.2.2

Combine the opposite terms in $0 + 0 + a h^{2} + k$ .

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + a h^{2} + k$

Step 4.2.2.2

Add $0$ and $a h^{2}$ .

$f (0) = a h^{2} + k$

Step 4.2.3

The final answer is $a h^{2} + k$ .

$a h^{2} + k$

Step 5

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

$(0, a h^{2} + k)$

Step 6