# Lesson 20

Evaluating Functions over Equal Intervals

- Let’s evaluate and rewrite expressions.

### 20.1: Finding Slopes

- Find the slope of each line.
- The line that passes through \((2,2)\) and \((3,6)\).
- The graph of \(f(x)=\text-2+\frac13x\).

- Show on the graph where each slope can be seen.

### 20.2: Incrementing by One

- For the function \(f(x)=3x+4\), evaluate:
- \(f(0)\) and \(f(1)\)
- \(f(100)\) and \(f(101)\)
- \(f(\text-10)\) and \(f(\text-9)\)
- \(f(0.5)\) and \(f(1.5)\)

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

- What do all those pairs of numbers you found have in common?
- Write an expression for \(g(u)\) and \(g(u+1)\).
- What would you expect to be the result of dividing \(g(u+1)\) by \(g(u)\)?
- 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

- Evaluate:
- \(\dfrac{3^5}{3^4}\)
- \(\dfrac{3^1}{3^0}\)
- \(\dfrac{3^{\text-1}}{3^{\text-2}}\)
- \(\dfrac{3^{100}}{3^{99}}\)
- \(\dfrac{3^{x+1}}{3^x}\)

- Solve for \(m\):
- \(\dfrac{2^m}{2^7}=2\)
- \(\dfrac{2^{100}}{2^m}=2\)
- \(\dfrac{2^m}{2^x}=2\)

- Write an equivalent expression using as few terms as possible:
- \(3(x+1) + 4 - (3x + 4)\)
- \(2(x+1) + 5 - (2x + 5)\)
- \(2(x+2) + 5 - (2(x+1) + 5)\)
- \(\text-5(x+1) + 3 - (\text-5x + 3)\)
- \(\dfrac{5^{x+1}}{5^x}\)
- \(\dfrac{7^{x+4}}{7^x}\)