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.
2 comma 2 and 3 comma 6 on a graph of a line 
Graph of line f. y intercept = -2. Slope = the fraction 1 over 3.

 

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. \(\dfrac{7^{x+4}}{7^x}\)

Summary