Comprehensive worksheet featuring eight problems designed to help students practice calculating the difference quotient for various types of functions.

Here are the step-by-step solutions for each problem on the worksheet.

The difference quotient formula is:
$$ \frac{f(x+h) - f(x)}{h} $$
*(Note: If the variable in the function is not $x$, like $b$ or $k$, we use that letter instead of $x$ in the formula.)*

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1. Find the difference quotient of $f(b) = -7b - 7$

Step 1: Find $f(b+h)$.
Replace every $b$ with $(b+h)$:
$$ f(b+h) = -7(b+h) - 7 $$
Distribute the $-7$:
$$ f(b+h) = -7b - 7h - 7 $$

Step 2: Subtract $f(b)$ from $f(b+h)$.
$$ f(b+h) - f(b) = (-7b - 7h - 7) - (-7b - 7) $$
$$ = -7b - 7h - 7 + 7b + 7 $$
Combine like terms ($-7b$ and $+7b$ cancel out; $-7$ and $+7$ cancel out):
$$ = -7h $$

Step 3: Divide by $h$.
$$ \frac{-7h}{h} = -7 $$

Final Answer:
-7

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2. Find the difference quotient of $f(k) = 4k - 5$

Step 1: Find $f(k+h)$.
$$ f(k+h) = 4(k+h) - 5 $$
$$ f(k+h) = 4k + 4h - 5 $$

Step 2: Subtract $f(k)$.
$$ (4k + 4h - 5) - (4k - 5) $$
$$ = 4k + 4h - 5 - 4k + 5 $$
Cancel terms ($4k$ and $-4k$; $-5$ and $+5$):
$$ = 4h $$

Step 3: Divide by $h$.
$$ \frac{4h}{h} = 4 $$

Final Answer:
4

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3. Find the difference quotient of $f(y) = y + 3$

Step 1: Find $f(y+h)$.
$$ f(y+h) = (y+h) + 3 = y + h + 3 $$

Step 2: Subtract $f(y)$.
$$ (y + h + 3) - (y + 3) $$
$$ = y + h + 3 - y - 3 $$
Cancel terms:
$$ = h $$

Step 3: Divide by $h$.
$$ \frac{h}{h} = 1 $$

Final Answer:
1

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4. Find the difference quotient of $f(c) = -2c - 9$

Step 1: Find $f(c+h)$.
$$ f(c+h) = -2(c+h) - 9 $$
$$ f(c+h) = -2c - 2h - 9 $$

Step 2: Subtract $f(c)$.
$$ (-2c - 2h - 9) - (-2c - 9) $$
$$ = -2c - 2h - 9 + 2c + 9 $$
Cancel terms:
$$ = -2h $$

Step 3: Divide by $h$.
$$ \frac{-2h}{h} = -2 $$

Final Answer:
-2

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5. Find the difference quotient of $f(c) = 6c^2 + 8c - 6$

Step 1: Find $f(c+h)$.
$$ f(c+h) = 6(c+h)^2 + 8(c+h) - 6 $$
Expand $(c+h)^2$ to $(c^2 + 2ch + h^2)$:
$$ = 6(c^2 + 2ch + h^2) + 8c + 8h - 6 $$
Distribute the 6:
$$ = 6c^2 + 12ch + 6h^2 + 8c + 8h - 6 $$

Step 2: Subtract $f(c)$.
$$ (6c^2 + 12ch + 6h^2 + 8c + 8h - 6) - (6c^2 + 8c - 6) $$
$$ = 6c^2 + 12ch + 6h^2 + 8c + 8h - 6 - 6c^2 - 8c + 6 $$
Cancel terms ($6c^2$, $8c$, $-6$):
$$ = 12ch + 6h^2 + 8h $$

Step 3: Divide by $h$.
Factor out an $h$ from the numerator:
$$ \frac{h(12c + 6h + 8)}{h} $$
Cancel the $h$:
$$ 12c + 6h + 8 $$

Final Answer:
12c + 6h + 8

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6. Find the difference quotient of $f(x) = 4x^2 + 6x + 6$

Step 1: Find $f(x+h)$.
$$ f(x+h) = 4(x+h)^2 + 6(x+h) + 6 $$
$$ = 4(x^2 + 2xh + h^2) + 6x + 6h + 6 $$
$$ = 4x^2 + 8xh + 4h^2 + 6x + 6h + 6 $$

Step 2: Subtract $f(x)$.
$$ (4x^2 + 8xh + 4h^2 + 6x + 6h + 6) - (4x^2 + 6x + 6) $$
Cancel terms ($4x^2$, $6x$, $6$):
$$ = 8xh + 4h^2 + 6h $$

Step 3: Divide by $h$.
$$ \frac{h(8x + 4h + 6)}{h} $$
$$ = 8x + 4h + 6 $$

Final Answer:
8x + 4h + 6

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7. Find the difference quotient of $f(x) = -10x^2 - 4x - 3$

Step 1: Find $f(x+h)$.
$$ f(x+h) = -10(x+h)^2 - 4(x+h) - 3 $$
$$ = -10(x^2 + 2xh + h^2) - 4x - 4h - 3 $$
$$ = -10x^2 - 20xh - 10h^2 - 4x - 4h - 3 $$

Step 2: Subtract $f(x)$.
$$ (-10x^2 - 20xh - 10h^2 - 4x - 4h - 3) - (-10x^2 - 4x - 3) $$
$$ = -10x^2 - 20xh - 10h^2 - 4x - 4h - 3 + 10x^2 + 4x + 3 $$
Cancel terms ($-10x^2$, $-4x$, $-3$):
$$ = -20xh - 10h^2 - 4h $$

Step 3: Divide by $h$.
$$ \frac{h(-20x - 10h - 4)}{h} $$
$$ = -20x - 10h - 4 $$

Final Answer:
-20x - 10h - 4

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8. Find the difference quotient of $f(m) = -\frac{3}{-2m+1}$

*Leave denominator in factored form.*

Step 1: Find $f(m+h)$.
$$ f(m+h) = -\frac{3}{-2(m+h)+1} = -\frac{3}{-2m - 2h + 1} $$

Step 2: Set up the subtraction $f(m+h) - f(m)$.
$$ -\frac{3}{-2m - 2h + 1} - \left( -\frac{3}{-2m + 1} \right) $$
This becomes addition because of the double negative:
$$ -\frac{3}{-2m - 2h + 1} + \frac{3}{-2m + 1} $$

To combine these fractions, find a common denominator: $(-2m - 2h + 1)(-2m + 1)$.

Numerator calculation:
$$ -3(-2m + 1) + 3(-2m - 2h + 1) $$
$$ = (6m - 3) + (-6m - 6h + 3) $$
$$ = 6m - 3 - 6m - 6h + 3 $$
Cancel terms ($6m$ and $-6m$; $-3$ and $3$):
$$ = -6h $$

So, $f(m+h) - f(m) = \frac{-6h}{(-2m - 2h + 1)(-2m + 1)}$

Step 3: Divide by $h$.
$$ \frac{\frac{-6h}{(-2m - 2h + 1)(-2m + 1)}}{h} $$
This is the same as multiplying by $\frac{1}{h}$:
$$ \frac{-6h}{h(-2m - 2h + 1)(-2m + 1)} $$
Cancel the $h$ on top and bottom:
$$ \frac{-6}{(-2m - 2h + 1)(-2m + 1)} $$

Final Answer:
$\frac{-6}{(-2m - 2h + 1)(-2m + 1)}$