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9.4 – Graphs of Exponential Functions

Key Terms

  • Horizontal Asymptote – A horizontal line that the graph of a function approaches but never intersects.
    • A function has a horizontal asymptote at every y-value where it is undefined and near which the function’s values become very large positive or negative numbers.
    • Think of an asymptote like an electric fence.  You can’t cross it!  You can’t even touch it!
  • Range – The set of a function’s output values (y-values).
  • y-intercept – A point where the graph of a function crosses the y-axis.
    • A function has at most one y-intercept.
    • The y-intercept of the line with the equation y = mx + b is the point (0, b).

Review

Graph Behavior
  • Increasing graphs slant or curve up from left to right
  • Decreasing graphs slant or curve down from left to right
  • All exponential graphs are curved, not straight.

Notes

The Exponential Function’s Base
  • The value of the base determines whether the graph increases or decreases from left to right.
  • The base for an exponential function can never be a negative number.
  • A negative value of b would make the function undefined for many values of x.

Alg1B 6.4 Review1

  • If b > 1, then the graph increases

Alg1B 6.4 Base Less than 1

  • If 0 < b < 1, then the graph decreases

Alg1B 6.4 Base Btwn 1 and 0

 

Intercepts, Points, and Asymptotes
  • The graph of F(x)=b^{x} will always have the y-intercept of (0, 1).
  • The graph of F(x)=b^{x} will always contain the point (1, b), where b is the base of the expression.
  • The x-axis is a horizontal asymptote: y = 0
  • Horizontal Asymptote: y = 0

Alg1B 6.4 Bounding Aymptote

  • Point (1, b) and y-intercept (0, 1)

Alg1B 6.4 Review Info

  • Example

Alg1B 6.4 Point 1 b

 

Domain and Range
  • The domain of a function is all the input values it will accept.
    • Domain ends with -in for input.

Alg1B 6.4 Domain

  • The range of a function is all the output values it will return.
    • Range starts with r for return.

Alg1B 6.4 Range

  • For a+b^{x}, the ranges is all positive real numbers greater than a.

 

(1, b) and (1, ab)
  • You can write F(x)=b^{x} in the form F(x)=a\bullet b^{x} by making a = 1.
  • The rule that the y-intercept = (0, a) and the other known point = (1, ab) is true for both forms of exponential functions.
  • The difference between the graphs below is that the y-intercept of the function F(x)=a\bullet b^{x} is the point (0, a), and its other known point is (1, ab).

Alg1B 6.4 a times b to the x

Alg1B 6.4 Compare

 

Vertical Translations (Shifting Up and Down)
  • Shifting affects the range, but not the domain.
    • The domain is always “All real numbers.”
    • Range will always be greater than the shift.
      • The range of the top graph is y > 5
      • The range of the bottom graph is y > -3

Alg2B 9.4 Translations

 

Horizontal Translations (Shifting Left and Right)
  • Horizontal shifts do not affect domain or range.
  • Horizontal shifts appear when a number is added to or subtracted from the input variable.
    • Ex. G(x)=2^{x-1} is shifted 1 unit to the right.
    • Ex. H(x)=2^{x+6} is shifted 6 units to the left.

Alg2B 9.4 Horizontal Shift

 

Vertical Stretching and Compressing
  • Multiplying a function by a negative number flips the graph across the x-axis.
  • When the function is flipped across the x-axis, the inequality sign of the range is reversed.

Alg2B 9.4 Compress Stretch

  • To horizontally stretch an exponential function, you multiply the input variable by a number between zero and 1 (a positive fraction).
  • To horizontally compress it, you multiply the input variable by a number greater than 1.

Alg2B 9.4 Compress Stretch 2

 

Translations (All Together)
Alg2B 9.4 Translate Ex

 

Exponential Functions and Graphs: Chart
Alg2B 9.4 Chart

Examples

  • Ex 1. The domain of the function given below is the set of all real numbers.
    • f(x)=(\frac{1}{7})^{x}
    • True: Any number can be an input.
  • Ex 2. What are the properties of F(x)=8\bullet 3^{x}:
    • The domain is all positive real numbers
    • The y-intercept is (0, 8)
    • The graph is increasing
  • Ex 3. The range of the function given below is the set of all positive real numbers greater than 9.
    • Answer: F(x)=9+3^{x}
  • Ex 4. The following facts are true for the graph of the function below
    • F(x)=(\frac{2}{7})^{x}
    • The range of F(x) is y > 0
    • The y-intercept is (0, 1)
    • It is decreasing
  • Ex 5. The following facts are true for the graph of the function below.
    • F(x)=7\bullet 5^{x}
    • The domain of F(x) is all real numbers
    • The y-intercept is (0, 7)
    • It is increasing
  • Ex 6. The graph below could be the graph of the following exponential function:
    • Answer: F(x)=3\bullet (1.2)^{x}

Alg1B 6.4 Qex6

  • Ex 7. The graph below could be the graph of the following exponential function:
    • Answer: F(x)=2\bullet (0.5)^{x}

Alg1B 6.4 Qex7


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