# Lesson 20

Evaluating Functions over Equal Intervals

• Let’s evaluate and rewrite expressions.

### 20.1: Finding Slopes

1. Find the slope of each line.
1. The line that passes through $$(2,2)$$ and $$(3,6)$$.
2. The graph of $$f(x)=\text-2+\frac13x$$.
2. Show on the graph where each slope can be seen.

### 20.2: Incrementing by One

1. For the function $$f(x)=3x+4$$, evaluate:
1. $$f(0)$$ and $$f(1)$$
2. $$f(100)$$ and $$f(101)$$
3. $$f(\text-10)$$ and $$f(\text-9)$$
4. $$f(0.5)$$ and $$f(1.5)$$
2. What do all those pairs of numbers you found have in common?
3. Write an expression for $$f(w)$$ and $$f(w+1)$$.
4. What would you expect to be the result of subtracting $$f(w)$$ from $$f(w+1)$$?
5. Subtract $$f(w)$$ from $$f(w+1)$$. If you don’t get the answer you predicted, work with a partner to check your algebra.
6. For the function $$g(x)=2^x$$, evaluate:
1. $$g(3)$$ and $$g(4)$$
2. $$g(0)$$ and $$g(1)$$
3. $$g(\text-1)$$ and $$g(\text-2)$$
4. $$g(10)$$ and $$g(11)$$
7. What do all those pairs of numbers you found have in common?
8. Write an expression for $$g(u)$$ and $$g(u+1)$$.
9. What would you expect to be the result of dividing $$g(u+1)$$ by $$g(u)$$?
10. Divide $$g(u+1)$$ by $$g(u)$$. If you don’t get the answer you predicted, work with a partner to check your algebra.

### 20.3: Rewriting Expressions

1. Evaluate:
1. $$\dfrac{3^5}{3^4}$$
2. $$\dfrac{3^1}{3^0}$$
3. $$\dfrac{3^{\text-1}}{3^{\text-2}}$$
4. $$\dfrac{3^{100}}{3^{99}}$$
5. $$\dfrac{3^{x+1}}{3^x}$$
2. Solve for $$m$$:
1. $$\dfrac{2^m}{2^7}=2$$
2. $$\dfrac{2^{100}}{2^m}=2$$
3. $$\dfrac{2^m}{2^x}=2$$
3. Write an equivalent expression using as few terms as possible:
1. $$3(x+1) + 4 - (3x + 4)$$
2. $$2(x+1) + 5 - (2x + 5)$$
3. $$2(x+2) + 5 - (2(x+1) + 5)$$
4. $$\text-5(x+1) + 3 - (\text-5x + 3)$$
5. $$\dfrac{5^{x+1}}{5^x}$$
6. $$\dfrac{7^{x+4}}{7^x}$$