Find @FORMULA f(x)=a|x-h|+k

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

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

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

Simplify the result.

Combine the opposite terms in $a | (x + h) - h | + k$ .

Subtract $h$ from $h$ .

$f (x + h) = a | x + 0 | + k$

Add $x$ and $0$ .

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

The final answer is $a | x | + k$ .

$a | x | + k$

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{a | x | + k - (a | x - h | + k)}{h}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{a | x | + k - (a | x - h |) - k}{h}$

Subtract $k$ from $k$ .

$\frac{a | x | - a | x - h | + 0}{h}$

Add $a | x | - a | x - h |$ and $0$ .

$\frac{a | x | - a | x - h |}{h}$

Rewrite $a | x | - a | x - h |$ in a factored form.

Factor $a$ out of $a | x | - a | x - h |$ .

Factor $a$ out of $a | x |$ .

$\frac{a (| x |) - a | x - h |}{h}$

Factor $a$ out of $- a | x - h |$ .

$\frac{a (| x |) + a (- 1 | x - h |)}{h}$

Factor $a$ out of $a (| x |) + a (- 1 | x - h |)$ .

$\frac{a (| x | - 1 | x - h |)}{h}$

Rewrite $- 1 | x - h |$ as $- | x - h |$ .

$\frac{a (| x | - | x - h |)}{h}$

Step 5