Graphing Exponential Functions Section 10.3 Graphing Exponential Functions

Graphing Exponential Functions with b > 1 Example Graph by hand. Solution List input–output pairs (see table) Input increases by 1 and output multiplies by 2 Plot these points (see next slide)

Graphing Exponential Functions with b > 1 Solution Continued Use graphing calculator to verify

Graphing Exponential Functions with 0< b < 1 Example Graph by hand. Solution List input–output pairs (see table) For example (–1, 8) is a solution x increases by 1, y is multiplied by ½

Graphing Exponential Functions with 0 < b < 1 Solution Continued

Property Illustration Base Multiplier Property; Increase or Decreasing Property Base Multiplier Property Property For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b. For the function , as the value of x increases by 1, the value of y is multiplied by 3 For the function , as the value of x increases by 1, the value of y is multiplied by 3/4 Illustration

Increase or Decrease Property Base Multiplier Property Property Let , where a > 0. Then If b > 1, then the function f is increasing. We say that the function grows exponentially (left). If 0 < b < 1, then the function f is decreasing. We say that the function decays exponentially (right).

y-intercept of an Exponential Function Intercepts Property For an exponential function of the form the y-intercept is (0, a). The function , the y-intercept is (0, 5) The function , the y-intercept is (0, 4) Illustration

Intercepts and Graph of an Exponential Function Warning Exponential function of the form , the y- intercept is not (0, b). By writing , we see that the y-intercept is (0, 1). For example, for , the y-intercept is (0, 1). Let 1. Find the y-intercept of f. Example

Intercepts and Graph of an Exponential Function Solution is of the form , We know that the y-intercept is (0, a), or (0, 6). 2. Find the x-intercept of f. By base multiplier property, x increases by 1, y value multiplies by ½ Example Solution

Intercepts and Graph of an Exponential Function Solution Continued No number of halvings will result in zero As x grows large, y gets closer to the x-axis Called horizontal asymptote 3. Graph f by hand. Example

Plot solutions from the table Intercepts and Graph of an Exponential Function Intercepts Solution Plot solutions from the table Verify on graphing calculator

Sketch and compare the graphs of . Graphs of Functions of the Form y=abx and y= –abx Reflection Property Example Sketch and compare the graphs of . Solution Input–output pairs are listed in the table g is a reflection of f across the x-axis

Graphs of Functions of the Form y=abx and y= –abx Reflection Property Example Find the domain and range of f. 5(3)x is defined for any real number x Domain: All real numbers Range: All positive real numbers Solution

Graphs of Functions of the Form y=abx and y= –abx Reflection Property Example Find the domain and range of g. – 5(3)x is defined for any real number x Domain: All real numbers Range: All negative real numbers Solution

Reflection Property Property Illustration The graphs of are reflections of each other across the x-axis. For all exponential functions the x-axis is a horizontal asymptote The range of an exponential function f(x) = abx is the set of all positive real numbers if a > 0, and the range is the set of all negative real number if a< 0. Illustration

Reflection Property Continued b > 1 (left) and 0 < b < 1 (right)

Finding Values of a Function from Its Graph Reflection Property Example The graph of an exponential function f is shown. Find f(2). Blue arrow shows input of x = 2 leads to an output y = 8 f(2) = 8 Solution

Finding Values of a Function from Its Graph Reflection Property Example 2. Find x when f(x) = 2. Red arrow shows output of y = 2 leads to an input x = -2 x = –2 when f(x) = 2 Solution

Finding Values of a Function from Its Graph Reflection Property Example 3. Find x when f(x) = 0. Graphs of exponential functions get close to zero, but never reaches x-axis No value of x where f(x) = 0 Solution