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Graphing exponential functions

MAT1510 - Precalculus Mathematics A · Exponential and Logarithmic Functions

Graphing Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a × b^x, where a is a constant, b is the base (a positive real number), and x is the exponent. The base b determines the growth or decay of the function. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.

Basic Characteristics of Exponential Functions

Exponential functions have several key characteristics:

  • The domain is all real numbers, x ∈ ℝ.
  • The range is positive real numbers, f(x) > 0.
  • The y-intercept occurs at (0, a), since f(0) = a × b^0 = a.
  • As x approaches negative infinity, f(x) approaches 0 (horizontal asymptote at y = 0).
  • As x approaches positive infinity, f(x) increases without bound (for b > 1) or decreases towards 0 (for 0 < b < 1).

Remember: The graph of an exponential function is always continuous and never touches the x-axis.

Graphing Exponential Growth Functions

To graph an exponential growth function, follow these steps:

  1. Identify the function in the form f(x) = a × b^x.
  2. Determine the y-intercept by calculating f(0).
  3. Choose additional x-values to calculate corresponding f(x) values.
  4. Plot the points on a Cartesian plane.
  5. Draw a smooth curve through the points, ensuring it approaches the horizontal asymptote.

Example 1: Graphing f(x) = 2 × 3^x

1. Identify the function: f(x) = 2 × 3^x.

2. Determine the y-intercept: f(0) = 2 × 3^0 = 2 × 1 = 2. The y-intercept is (0, 2).

3. Choose additional x-values: Let’s choose x = -1, 1, and 2.

  • For x = -1: f(-1) = 2 × 3^(-1) = 2 × (1/3) = 2/3.
  • For x = 1: f(1) = 2 × 3^1 = 2 × 3 = 6.
  • For x = 2: f(2) = 2 × 3^2 = 2 × 9 = 18.

4. The points to plot are: (0, 2), (-1, 2/3), (1, 6), and (2, 18).

5. Plot these points on the Cartesian plane and draw a smooth curve through them, approaching the x-axis as x decreases.

Graphing Exponential Decay Functions

To graph an exponential decay function, the steps are similar:

  1. Identify the function in the form f(x) = a × b^x, where 0 < b < 1.
  2. Determine the y-intercept by calculating f(0).
  3. Choose additional x-values to calculate corresponding f(x) values.
  4. Plot the points on a Cartesian plane.
  5. Draw a smooth curve through the points, ensuring it approaches the horizontal asymptote.

Example 2: Graphing f(x) = 5 × (1/2)^x

1. Identify the function: f(x) = 5 × (1/2)^x.

2. Determine the y-intercept: f(0) = 5 × (1/2)^0 = 5 × 1 = 5. The y-intercept is (0, 5).

3. Choose additional x-values: Let’s choose x = -1, 1, and 2.

  • For x = -1: f(-1) = 5 × (1/2)^(-1) = 5 × 2 = 10.
  • For x = 1: f(1) = 5 × (1/2)^1 = 5 × (1/2) = 2.5.
  • For x = 2: f(2) = 5 × (1/2)^2 = 5 × (1/4) = 1.25.

4. The points to plot are: (0, 5), (-1, 10), (1, 2.5), and (2, 1.25).

5. Plot these points on the Cartesian plane and draw a smooth curve through them, approaching the x-axis as x increases.

Watch out: Remember that for exponential decay, the function will decrease towards the horizontal asymptote but never touch it.

Transformations of Exponential Functions

Exponential functions can be transformed by shifting, reflecting, or stretching. The general form is:

f(x) = a × b^(x - h) + k

where (h, k) is the point of transformation. Here, h represents a horizontal shift and k represents a vertical shift.

Example 3: Graphing f(x) = 2 × 3^(x - 1) + 1

1. Identify the function: f(x) = 2 × 3^(x - 1) + 1.

2. Determine the y-intercept: f(0) = 2 × 3^(0 - 1) + 1 = 2 × (1/3) + 1 = 2/3 + 1 = 5/3. The y-intercept is (0, 5/3).

3. Choose additional x-values: Let’s choose x = 0, 1, and 2.

  • For x = 0: f(0) = 2 × 3^(0 - 1) + 1 = 5/3.
  • For x = 1: f(1) = 2 × 3^(1 - 1) + 1 = 2 × 1 + 1 = 3.
  • For x = 2: f(2) = 2 × 3^(2 - 1) + 1 = 2 × 3 + 1 = 7.

4. The points to plot are: (0, 5/3), (1, 3), and (2, 7).

5. Plot these points on the Cartesian plane and draw a smooth curve through them, ensuring it reflects the transformation.

Tip: When graphing transformations, always consider the shifts and how they affect the position of the graph.

Summary

  • Exponential functions can represent growth or decay.
  • The graph of an exponential function is continuous and has a horizontal asymptote.
  • Transformations can shift, reflect, or stretch the graph.

Check your understanding

  1. What is the general form of an exponential function?
  2. How do you determine the y-intercept of an exponential function?
  3. What happens to the graph of an exponential decay function as x increases?
  4. Describe how a transformation affects the graph of an exponential function.