Find @FORMULA f(x)=3x^2-7

$f (x) = 3 x^{2} - 7$

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.

$f (x + h) = 3 {(x + h)}^{2} - 7$

Simplify the result.

Simplify each term.

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

$f (x + h) = 3 ((x + h) (x + h)) - 7$

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

Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h)) - 7$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h)) - 7$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h) - 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 3 (x^{2} + x h + h x + h \cdot h) - 7$

Multiply $h$ by $h$ .

$f (x + h) = 3 (x^{2} + x h + h x + h^{2}) - 7$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 3 (x^{2} + h x + h x + h^{2}) - 7$

Add $h x$ and $h x$ .

$f (x + h) = 3 (x^{2} + 2 h x + h^{2}) - 7$

Apply the distributive property.

$f (x + h) = 3 x^{2} + 3 (2 h x) + 3 h^{2} - 7$

Multiply $2$ by $3$ .

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} - 7$

The final answer is $3 x^{2} + 6 h x + 3 h^{2} - 7$ .

$3 x^{2} + 6 h x + 3 h^{2} - 7$

Reorder.

Move $3 x^{2}$ .

$6 h x + 3 h^{2} + 3 x^{2} - 7$

Reorder $6 h x$ and $3 h^{2}$ .

$3 h^{2} + 6 h x + 3 x^{2} - 7$

Find the components of the definition.

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} - 7$

$f (x) = 3 x^{2} - 7$

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} - 7$

$f (x) = 3 x^{2} - 7$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{3 h^{2} + 6 h x + 3 x^{2} - 7 - (3 x^{2} - 7)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{3 h^{2} + 6 h x + 3 x^{2} - 7 - (3 x^{2}) - - 7}{h}$

Multiply $3$ by $- 1$ .

$\frac{3 h^{2} + 6 h x + 3 x^{2} - 7 - 3 x^{2} - - 7}{h}$

Multiply $- 1$ by $- 7$ .

$\frac{3 h^{2} + 6 h x + 3 x^{2} - 7 - 3 x^{2} + 7}{h}$

Subtract $3 x^{2}$ from $3 x^{2}$ .

$\frac{3 h^{2} + 6 h x + 0 - 7 + 7}{h}$

Add $3 h^{2}$ and $0$ .

$\frac{3 h^{2} + 6 h x - 7 + 7}{h}$

Add $- 7$ and $7$ .

$\frac{3 h^{2} + 6 h x + 0}{h}$

Add $3 h^{2} + 6 h x$ and $0$ .

$\frac{3 h^{2} + 6 h x}{h}$

Factor $3 h$ out of $3 h^{2} + 6 h x$ .

Factor $3 h$ out of $3 h^{2}$ .

$\frac{3 h \cdot h + 6 h x}{h}$

Factor $3 h$ out of $6 h x$ .

$\frac{3 h \cdot h + 3 h (2 x)}{h}$

Factor $3 h$ out of $3 h \cdot h + 3 h (2 x)$ .

$\frac{3 h (h + 2 x)}{h}$

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{3 h (h + 2 x)}{h}$

Divide $3 (h + 2 x)$ by $1$ .

$3 (h + 2 x)$

Apply the distributive property.

$3 h + 3 (2 x)$

Simplify the expression.

Multiply $2$ by $3$ .

$3 h + 6 x$

Reorder $3 h$ and $6 x$ .

$6 x + 3 h$

Step 5

Find @FORMULA f(x)=3x^2-7

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