Linear vs Exponential

Grade 9 · algebra · 97 practice problems · read aloud

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Linear vs. Exponential Growth

Understanding the difference between linear and exponential growth is crucial for predicting how quantities change over time, from saving money to understanding population growth.

What's the Difference? 🤔

Linear Growth increases by a constant amount over time. The graph is a straight line.

Exponential Growth increases by a constant percentage or multiplier over time. The graph is a curve that gets steeper and steeper.

How to Tell Them Apart: A Step-by-Step Guide

  1. Look at the Pattern: Is the quantity growing by adding the same number (linear) or by multiplying by the same number (exponential)?
  2. Check the Equation:
    • Linear: y = mx + b (e.g., y = 5x + 3)
    • Exponential: y = a ⋅ bˣ (e.g., y = 3 ⋅ 2ˣ)
  3. Ask the Question: "Am I adding the same amount each time, or am I multiplying by the same amount?"

Visual Examples

Example 1: Linear Growth

You get a $10 weekly allowance. How much do you have after 4 weeks?

Pattern: Start: $10 → Week 1: $20 → Week 2: $30 → Week 3: $40

You add $10 each week. This is linear (y = 10x + 10).

Example 2: Exponential Growth

A rumor spreads so that each person tells 2 new people. If 1 person starts, how many know after 3 rounds?

Pattern: Round 0: 1 → Round 1: 2 → Round 2: 4 → Round 3: 8

You multiply by 2 each round. This is exponential (y = 1 ⋅ 2ˣ).

Common Mistakes to Avoid 🚫

Mistake: Thinking a large constant addition is exponential.

Reality: Even adding 1000 each time is still linear. Exponential is about multiplication.

Mistake: Confusing the exponent in an exponential equation.

Reality: In y = 2ˣ, the variable (x) is in the exponent. In y = x², it's not; that's quadratic growth.

Tips & Tricks

Word Clues: "Per," "each," "every" often signal linear (constant rate). "Doubles," "triples," "percent increase" often signal exponential.

The Table Test: Make a table of values. If the y-values have a common difference (you subtract to get the same number), it's linear. If they have a common ratio (you divide to get the same number), it's exponential.

How to Practice

  • Create your own real-world scenarios (e.g., phone plans with a flat fee vs. a compounding data plan).
  • Use graphing calculators or online tools to plot linear (y=2x+1) and exponential (y=2ˣ) functions side-by-side.
  • Practice sorting word problems into "Linear" and "Exponential" piles.

Practice problems

6 of the 97, worked through step by step — try them before opening the answer.

1 2^(x+1) = 16

Hint: Express both sides with the same base to solve for the variable.

Show the answer

Answer: 3

  1. Recognize that 16 can be written as a power of 2. 16 = 2 * 2 * 2 * 2 = 2^4. So we rewrite the equation as: 2^(x+1) = 2^4
  2. Since the bases are the same (base 2), we can set the exponents equal to each other. x + 1 = 4
  3. Solve for x. x = 4 - 1 x = 3
  4. Check the solution. 2^(3+1) = 2^4 = 16, which matches the original equation. Thus,

We start with the equation: 2^(x+1) = 16 the correct answer is x = 3.

2 2^(x-1) = 32

Hint: Express both sides with the same base to solve for the exponent

Show the answer

Answer: 6

  1. Recognize that 32 can be written as a power of 2
  2. 32 = 2^5
  3. Rewrite the equation: 2^(x-1) = 2^5
  4. Since the bases are equal, set the exponents equal: x - 1 = 5
  5. Solve for x: x = 5 + 1
  6. x = 6

The answer is 6.

3 2^(x+3) = 32

Hint: Express both sides with the same base to solve for the exponent

Show the answer

Answer: 2

  1. Write 32 as a power of 2: 32 = 2^5
  2. Substitute into the equation: 2^(x+3) = 2^5
  3. Since the bases are equal, set the exponents equal: x + 3 = 5
  4. Solve for x: x = 5 - 3
  5. x = 2

The answer is 2.

4 2^(x) = 32, x = ?

Hint: Consider what power you need to raise 2 to in order to get the result. Try expressing the number on the right as a power of 2.

Show the answer

Answer: 5

  1. Write the equation: 2^x = 32
  2. Express 32 as a power of 2: 32 = 2^5
  3. Substitute back into the equation: 2^x = 2^5
  4. Since the bases are equal, the exponents must be equal: x = 5
  5. Verify: 2^5 = 32, which matches the original equation

The answer is 5.

5 f(x) = 2^x, f(4) = ?

Hint: In exponential functions, the base is raised to the power of the input value. For example, if g(x) = 3^x, then g(2) would be 3 raised to the second power.

Show the answer

Answer: 16

  1. Substitute x = 4 into the function. f(4) = 2^4
  2. Understand that 2^4 means 2 multiplied by itself 4 times. 2^4 = 2 * 2 * 2 * 2
  3. Perform the multiplication step by step. First, 2 * 2 = 4 Then, 4 * 2 = 8 Then, 8 * 2 = 16
  4. Conclude the result. f(4) = 16 Final answer: 16

We are given the function: f(x) = 2^x We are asked to find f(4).

6 f(x) = 2^x, f(3) = ?

Hint: For exponential functions, substitute the input value into the exponent position and evaluate the power.

Show the answer

Answer: 8

  1. Substitute x = 3 into the function. f(3) = 2^3
  2. Understand that 2^3 means 2 multiplied by itself 3 times. 2^3 = 2 * 2 * 2
  3. Perform the multiplication step by step. First, 2 * 2 = 4 Then, 4 * 2 = 8
  4. Write the final answer. f(3) = 8

We are given the function f(x) = 2^x and asked to find f(3).

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