Find the Difference Quotient f(x)=x^2-4x+1

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

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.

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

Simplify the result.

Simplify each term.

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

$f (x + h) = (x + h) (x + h) - 4 (x + h) + 1$

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

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) - 4 (x + h) + 1$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) - 4 (x + h) + 1$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h - 4 (x + h) + 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

The final answer is $x^{2} + 2 h x + h^{2} - 4 x - 4 h + 1$ .

$x^{2} + 2 h x + h^{2} - 4 x - 4 h + 1$

Reorder.

Move $- 4 x$ .

$x^{2} + 2 h x + h^{2} - 4 h - 4 x + 1$

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} - 4 h - 4 x + 1$

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

$h^{2} + 2 h x + x^{2} - 4 h - 4 x + 1$

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Simplify.

Multiply $- 4$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

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

$\frac{h^{2} + 2 h x - 4 h - 4 x + 1 + 0 + 4 x - 1}{h}$

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

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

Add $- 4 x$ and $4 x$ .

$\frac{h^{2} + 2 h x - 4 h + 0 + 1 - 1}{h}$

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

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

Subtract $1$ from $1$ .

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

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

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

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

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

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

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

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

Factor $h$ out of $- 4 h$ .

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

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

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

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

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

$h + 2 x - 4$

Reorder $h$ and $2 x$ .

$2 x + h - 4$

Step 5

Find the Difference Quotient f(x)=x^2-4x+1

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