Precalculus Examples

$y = {(x + 2)}^{2}$

Step 1

Write $y = {(x + 2)}^{2}$ as a function.

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

Step 2

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 3

Find the components of the definition.

Tap for more steps...

Step 3.1

Evaluate the function at $x = x + h$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

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

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

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

$f (x + h) = (x + h + 2) (x + h + 2)$

Step 3.1.2.2

Expand $(x + h + 2) (x + h + 2)$ by multiplying each term in the first expression by each term in the second expression.

$f (x + h) = x \cdot x + x h + x \cdot 2 + h x + h \cdot h + h \cdot 2 + 2 x + 2 h + 2 \cdot 2$

Step 3.1.2.3

Simplify each term.

Tap for more steps...

Step 3.1.2.3.1

Multiply $x$ by $x$ .

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

Step 3.1.2.3.2

Move $2$ to the left of $x$ .

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

Step 3.1.2.3.3

Multiply $h$ by $h$ .

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

Step 3.1.2.3.4

Move $2$ to the left of $h$ .

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

Step 3.1.2.3.5

Multiply $2$ by $2$ .

$f (x + h) = x^{2} + x h + 2 x + h x + h^{2} + 2 h + 2 x + 2 h + 4$

Step 3.1.2.4

Add $x h$ and $h x$ .

Tap for more steps...

Step 3.1.2.4.1

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + 2 x + h^{2} + 2 h + 2 x + 2 h + 4$

Step 3.1.2.4.2

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + 2 x + h^{2} + 2 h + 2 x + 2 h + 4$

Step 3.1.2.5

Add $2 x$ and $2 x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 2 h + 4 x + 2 h + 4$

Step 3.1.2.6

Add $2 h$ and $2 h$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 4 h + 4 x + 4$

Step 3.1.2.7

The final answer is $x^{2} + 2 h x + h^{2} + 4 h + 4 x + 4$ .

$x^{2} + 2 h x + h^{2} + 4 h + 4 x + 4$

Step 3.2

Reorder.

Tap for more steps...

Step 3.2.1

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} + 4 h + 4 x + 4$

Step 3.2.2

Reorder $2 h x$ and $h^{2}$ .

$h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4$

Step 3.3

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4$

$f (x) = x^{2} + 4 x + 4$

$f (x + h) = h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4$

$f (x) = x^{2} + 4 x + 4$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4 - (x^{2} + 4 x + 4)}{h}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Apply the distributive property.

$\frac{h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4 - x^{2} - (4 x) - 1 \cdot 4}{h}$

Step 5.1.2

Simplify.

Tap for more steps...

Step 5.1.2.1

Multiply $4$ by $- 1$ .

$\frac{h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4 - x^{2} - 4 x - 1 \cdot 4}{h}$

Step 5.1.2.2

Multiply $- 1$ by $4$ .

$\frac{h^{2} + 2 h x + x^{2} + 4 h + 4 x + 4 - x^{2} - 4 x - 4}{h}$

Step 5.1.3

Subtract $x^{2}$ from $x^{2}$ .

$\frac{h^{2} + 2 h x + 4 h + 4 x + 4 + 0 - 4 x - 4}{h}$

Step 5.1.4

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

$\frac{h^{2} + 2 h x + 4 h + 4 x + 4 - 4 x - 4}{h}$

Step 5.1.5

Subtract $4 x$ from $4 x$ .

$\frac{h^{2} + 2 h x + 4 h + 0 + 4 - 4}{h}$

Step 5.1.6

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

$\frac{h^{2} + 2 h x + 4 h + 4 - 4}{h}$

Step 5.1.7

Subtract $4$ from $4$ .

$\frac{h^{2} + 2 h x + 4 h + 0}{h}$

Step 5.1.8

Add $h^{2} + 2 h x + 4 h$ and $0$ .

$\frac{h^{2} + 2 h x + 4 h}{h}$

Step 5.1.9

Factor $h$ out of $h^{2} + 2 h x + 4 h$ .

Tap for more steps...

Step 5.1.9.1

Factor $h$ out of $h^{2}$ .

$\frac{h \cdot h + 2 h x + 4 h}{h}$

Step 5.1.9.2

Factor $h$ out of $2 h x$ .

$\frac{h (h) + h (2 x) + 4 h}{h}$

Step 5.1.9.3

Factor $h$ out of $4 h$ .

$\frac{h (h) + h (2 x) + h \cdot 4}{h}$

Step 5.1.9.4

Factor $h$ out of $h (h) + h (2 x)$ .

$\frac{h (h + 2 x) + h \cdot 4}{h}$

Step 5.1.9.5

Factor $h$ out of $h (h + 2 x) + h \cdot 4$ .

$\frac{h (h + 2 x + 4)}{h}$

Step 5.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $h$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{h (h + 2 x + 4)}{h}$

Step 5.2.1.2

Divide $h + 2 x + 4$ by $1$ .

$h + 2 x + 4$

Step 5.2.2

Reorder $h$ and $2 x$ .

$2 x + h + 4$

Step 6