Calculate the difference quotient Worksheet Name: Date: $$ \qquad $$ Period: $$ \qquad $$ 1. Find the difference quotient of the function below. $$ f(b)=-7 b-7 $$ 2. Find the difference quotient of the function below. $$ f(k)=4 k-5 $$ 4. Find the difference quotient of the function below. $$ f(c)=-2 c-9 $$ 6. Find the difference quotient of the function below. $$ f(x)=4 x^{2}+6 x+6 $$ 8. Find the difference quotient, $$ \frac{f(m+h)-f(m)}{(m+h)-m} $$, of the function below. In your answer, leave your denominator in factored form. $$ f(m)=-\frac{3}{(-2 m+1)} $$

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1. For the function $$ f(b)=-7b-7 $$:
   
   - Compute $$ f(b+h) $$:
     $$
     f(b+h) = -7(b+h)-7 = -7b-7h-7
     $$
   - Find the difference quotient:
     $$
     \frac{f(b+h)-f(b)}{h} = \frac{(-7b-7h-7)-(-7b-7)}{h} = \frac{-7h}{h} = -7
     $$
     
2. For the function $$ f(k)=4k-5 $$:
   
   - Compute $$ f(k+h) $$:
     $$
     f(k+h) = 4(k+h)-5 = 4k+4h-5
     $$
   - Find the difference quotient:
     $$
     \frac{f(k+h)-f(k)}{h} = \frac{(4k+4h-5)-(4k-5)}{h} = \frac{4h}{h} = 4
     $$

4. For the function $$ f(c)=-2c-9 $$:
   
   - Compute $$ f(c+h) $$:
     $$
     f(c+h) = -2(c+h)-9 = -2c-2h-9
     $$
   - Find the difference quotient:
     $$
     \frac{f(c+h)-f(c)}{h} = \frac{(-2c-2h-9)-(-2c-9)}{h} = \frac{-2h}{h} = -2
     $$

6. For the function $$ f(x)=4x^{2}+6x+6 $$:
   
   - Compute $$ f(x+h) $$:
     $$
     f(x+h)=4(x+h)^2+6(x+h)+6 = 4(x^2+2xh+h^2)+6x+6h+6 = 4x^2+8xh+4h^2+6x+6h+6
     $$
   - Find the difference quotient:
     $$
     \frac{f(x+h)-f(x)}{h} = \frac{(4x^2+8xh+4h^2+6x+6h+6)-(4x^2+6x+6)}{h} = \frac{8xh+4h^2+6h}{h}
     $$
     Factor out $$ h $$:
     $$
     \frac{h(8x+4h+6)}{h} = 8x+4h+6
     $$

8. For the function $$ f(m)=-\frac{3}{-2m+1} $$:
   
   - Write $$ f(m) $$ and $$ f(m+h) $$ in a more convenient form:
     $$
     f(m) = -\frac{3}{-2m+1} = -\frac{3}{1-2m}, \quad f(m+h) = -\frac{3}{-2(m+h)+1} = -\frac{3}{1-2m-2h}
     $$
   - Compute the difference quotient:
     $$
     \frac{f(m+h)-f(m)}{h} = \frac{-\frac{3}{1-2m-2h} - \left(-\frac{3}{1-2m}\right)}{h} = \frac{-\frac{3}{1-2m-2h}+\frac{3}{1-2m}}{h}
     $$
   - Combine the fractions in the numerator:
     $$
     -\frac{3}{1-2m-2h}+\frac{3}{1-2m} = 3\left(\frac{1}{1-2m} - \frac{1}{1-2m-2h}\right)
     $$
   - Find a common denominator:
     $$
     \frac{1}{1-2m}-\frac{1}{1-2m-2h} = \frac{(1-2m-2h)-(1-2m)}{(1-2m)(1-2m-2h)} = \frac{-2h}{(1-2m)(1-2m-2h)}
     $$
   - Hence, the numerator becomes:
     $$
     3\left(\frac{-2h}{(1-2m)(1-2m-2h)}\right) = \frac{-6h}{(1-2m)(1-2m-2h)}
     $$
   - Dividing by $$ h $$ we get:
     $$
     \frac{-6h}{(1-2m)(1-2m-2h)} \cdot \frac{1}{h} = \frac{-6}{(1-2m)(1-2m-2h)}
     $$
   - The denominator is already in factored form.

Thus, the difference quotient for each function is:

1. $$ -7 $$

2. $$ 4 $$

4. $$ -2 $$

6. $$ 8x+4h+6 $$

8. $$ \displaystyle \frac{-6}{(1-2m)(1-2m-2h)} $$
1. The difference quotient for $$ f(b) = -7b - 7 $$ is **-7**. 2. The difference quotient for $$ f(k) = 4k - 5 $$ is **4**. 4. The difference quotient for $$ f(c) = -2c - 9 $$ is **-2**. 6. The difference quotient for $$ f(x) = 4x^{2} + 6x + 6 $$ is **$$8x + 4h + 6$$**. 8. The difference quotient for $$ f(m) = -\frac{3}{-2m + 1} $$ is **$$\frac{-6}{(1 - 2m)(1 - 2m - 2h)}$$**.

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