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Algebra 1 Unit 8 Lesson 2: Writing Exponential Functions

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Summary of “Algebra 1 Unit 8 Lesson 2: Writing Exponential Functions”

This lesson focuses on how to write exponential functions from given tables or graphs. The key goal is to express the function in the standard exponential form:

[ y = a \times b^x ]

where:

  • (a) is the y-intercept (initial value when (x=0)),
  • (b) is the scale factor or common ratio (also called the growth or decay factor),
  • (x) is the exponent (independent variable).

Main Concepts and Lessons

  • Exponential Function Form: The general form is (y = a \times b^x).

    • (a) represents the y-intercept or starting value.
    • (b) is the common ratio or scale factor showing how the function grows or decays.

  • Finding (a) (y-intercept): Identify the value of (y) when (x=0) from the table or graph.

  • Finding (b) (scale factor/common ratio):

    • Calculate by dividing a (y)-value by the previous (y)-value in the table or graph.
    • Use multiple pairs of points to confirm (b) is consistent.
    • If (b > 1), the function represents exponential growth.
    • If (0 < b < 1), the function represents exponential decay.

  • Interpreting Growth and Decay:

    • Growth: (b > 1), values increase as (x) increases.
    • Decay: (0 < b < 1), values decrease as (x) increases.

  • Writing the Equation:

    • Once (a) and (b) are found, plug them into (y = a \times b^x).
    • (b) can be expressed as a fraction or decimal (e.g., (1/3) or 0.33). Both forms are acceptable.

  • Using Graphs:

    • Identify the y-intercept visually.
    • Select points with clear (y)-values to calculate the ratio (b).
    • Confirm whether the function is growth or decay based on (b).

  • Examples from the Lesson:

    • Example 1: (y = 1 \times 2^x) (growth, doubling each step).
    • Example 2: (y = 16 \times 2^x) (growth).
    • Example 3: (y = 6 \times (1/3)^x) (decay).
    • Example 4: (y = 25 \times (1/2)^x) (decay).
    • Graph examples showing similar calculations.

  • Interpreting Exponential Functions in Real-World Contexts:

    • (a) represents the starting amount (e.g., initial money in an account).
    • (b) represents the growth or decay factor.
      • To find the percentage growth: ((b - 1) \times 100\%).
      • To find the percentage decay: ((1 - b) \times 100\%).
    • (x) (or the exponent) represents time or another independent variable.

  • Worked Examples with Money:

    • (A = 700 \times 1.09^t): 700 is the initial amount, 1.09 is the growth factor (9% increase).
    • (A = 250 \times 0.75^t): 250 is the initial amount, 0.75 is the decay factor (25% decrease).

Methodology / Step-by-Step Instructions to Write Exponential Functions from a Table or Graph

  1. Identify the y-intercept ((a)): Find the (y) value when (x = 0).

  2. Calculate the common ratio ((b)): Divide a (y)-value by the previous (y)-value. Check multiple pairs to confirm the ratio is consistent.

  3. Determine if the function is growth or decay:

    • If (b > 1), it is growth.
    • If (0 < b < 1), it is decay.

  4. Write the exponential function: Use the formula (y = a \times b^x). Express (b) as a fraction or decimal.

  5. Interpret the function (if applicable): For real-world problems, interpret (a), (b), and (x). Calculate percentage growth or decay by converting (b) accordingly.

  6. Practice: Use the above steps on provided tables or graphs. Confirm results by checking multiple points.

Speakers / Sources Featured

  • The lesson is delivered by a single instructor (unnamed) who explains concepts, works through examples, and guides students through practice problems.
  • No other speakers or sources are explicitly mentioned.

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