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6.4 – Graphs of Exponential Functions

Key Terms

  • x- axis – The horizontal axis on the Cartesian coordinate system.
  • 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!

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

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

 

Exponential Functions and Graphs: Chart
Alg1B 6.4 Graph Details2

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. The range of F(x)=8\bullet 3^{x} is:
    • Answer: All positive real numbers (this is a rule).
  • 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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