Let’s check your mastery! Be the master of Exponential Functions Determine whether the exponential function is a growth or decay. f(x) = 𝟑 𝒙 ( ) 𝟓 f(x) = 7x f(x) = 𝟖 𝒙 ( ) 𝟑 Exponential Decay f(x) = 0.0205x Exponential Growth Exponential Decay Exponential Growth f(x) = ( 𝟏𝟔)𝒙 Exponential Growth f(x) = 1.005x Exponential Growth Determine whether the exponential function is a growth or decay. Exponential Decay Exponential Growth Think of a number between 1 and 10. Multiply it by 9 and subtract 1. Now close your eyes. It’s dark isn’t it! Happy Friday everyone! Lesson 2 Graph of Exponential Function continuation Exponential Function of the x Form f(x) = ab + k The Vertical Shift Consider the graphs of exponential functions below What type of Exponential Function is being represented by the graph? Exponential Growth Whatare are characteristics What thethe characteristics or or ++=6? features of of thethe graph f(x) =f(x) 2x +2? 3? features graph 2x ? Domain: All REAL NUMBERS, (-∞, ∞) Range: yy > 3 >6 2 0 y-intercept: x-intercept: Asymptote: f(x) = 2x + 2 f(x) = 2x f(x) = 2x + 3 (0,7) (0,4) (0,3) (0,1) f(x) = 2x + 6 No x-intercept: yy == 620 3 What features features of of the the original original graph graph are are What affected when when we we translated translated the the graph? graph? affected f(x) =2x f(x) = abx + k a = 1, b = 2, k = 0 Range: y>0 f(x) =2x+ 2 f(x) = abx + k a = 1, b = 2, k = 2 y>2 f(x) =2x+ 3 f(x) =2x+ 6 f(x) = abx + k f(x) = abx + k a = 1, b = 2, k = 3 a = 1, b = 2, k = 6 y>3 y>6 y-intercept: (0,1) (0,3) (0,4) (0,7) Asymptote: y=0 y=2 y=3 y=6 Given an exponential function, can we determine the characteristics without actually graphing it? Consider the graphs of exponential functions below What type of Exponential Function is being represented by the graph? f(x) = 2x What arethe thecharacteristics characteristics What are or or features ofofthe 8? graph f(x)f(x) = 2x=- 4? features the graph 2x ? Domain: All REAL NUMBERS, (-∞, ∞) Range: 4 yy > >0--8 y-intercept: -7) (0, (0,1) -3) x-intercept: (2, No 0) x-intercept (3, Asymptote: y = -0-84 What features of the original graph are affected when we translated the graph? f(x) = 2x - 4 f(x) = 2x - 8 f(x) =2x f(x) = abx + k a = 1, b = 2, k = 0 Range: y-intercept: x-intercept: Asymptote: f(x) =2x - 4 + k f(x) = abx + k a = 1, b = 2, k = - 4 a = 1, b = 2, k = -8 f(x) = y>0 (0,1) No x-intercept y=0 f(x) =2x - 8 abx y>−4 y>- 8 (0,- 3) (0, - 7) (2,0) (3, 0) y=-4 y = -8 Given an exponential function, can we determine the characteristics without actually graphing it? Remember: Graphing an Exponential Function with a Vertical Shift An exponential function of the form f(x) = abx+ k is an exponential function with a vertical shift. The constant k is what causes the vertical shift to occur. A vertical shift is when the graph of the function is moved up or down a fixed distance, k. When a vertical shift is applied to an exponential function, what features of the graph are affected? The features of the graph that are affected are the intercepts, horizontal asymptote, and range. The y-intercept will move up or down a fixed amount, k, and the horizontal asymptote will also move up or down a fixed amount, k. Moving the horizontal asymptote up or down will then change the range of the function because the graph cannot touch or go below the horizontal asymptote. Exponential Function of the x-h Form f(x) = ab The Horizontal Shift Consider the graphs of exponential functions below What type of Exponential Function is being represented by the graph? What What are are the the characteristics characteristics or or 𝟏𝟏𝒙+𝟐 𝒙−𝟑 features of the graph f(x) = ( f(x) ==((𝟐)𝒙)? ?? features of the graph f(x) 𝟏 f(x) = (𝟐)𝒙−𝟑 𝟐 Domain: All REAL NUMBERS, (-∞, ∞) Range: y>0 𝟏 f(x) = (𝟐)𝒙 y-intercept: (0,1) (0, (0,8) 1/4 ) x-intercept: No x-intercept: Asymptote: 𝟏 𝟐 f(x) = ( )𝒙+𝟐 y=0 What features of the original graph are affected when we translated the graph? f(x) = 𝟏 𝒙 ( ) 𝟐 𝒙−𝒉 f(x) = 𝒂𝒃𝒙−𝒉 a = 1, b = 𝟏 ( ), 𝟐 Range: y-intercept: x-intercept: Asymptote: f(x) = 𝟏 𝒙+𝟐 ( ) 𝟐 f(x) = 𝒂𝒃 h= 0 a = 1, b = y>0 (0,1) No x-intercept y=0 𝟏 ( ), 𝟐 h = -2 (0, f(x) = 𝒂𝒃𝒙−𝒉 a = 1, b = 𝟏 ( ), 𝟐 h=3 y>0 y>0 𝟏 ( ) 𝟒 f(x) = 𝟏 𝒙−𝟑 ( ) 𝟐 ) (0, 8) No x-intercept No x-intercept y=0 y=0 Given an exponential function, can we determine the characteristics without actually graphing it? Remember: Graphing an Exponential Horizontal Shift Function with a An exponential function of the form f(x) = ab x – h is an exponential function with a horizontal shift. The constant h is what causes the horizontal shift to occur. A horizontal shift is when the graph of the function is moved to the left or right a fixed distance, h. When a horizontal shift is applied to an exponential function, what features of the graph are affected? The only feature of the graph that is affected when a = 1 is the y-intercept. The y-intercept will move to the left or right a fixed amount, h. This new point will NO longer be the y-intercept, but it will be a point on the graph that we can use as a reference point when visualizing the graph. The horizontal asymptote will still be at the x-axis or y = 0. Exponential Function of the Form f(x) x-h = ab + k xab Exponential Function of the Form f(x) = h + k have both a vertical shift of k units and a horizontal shift of h units. Clue : h – horizontal shift Therefore k - vertical shift Consider the f(x) = abx-h + k where a = 1 and b > 1 f(x) = x-1 2 -2 Exponential Growth 2x-1 – 2 of the y x Characteristics graph 15 (-2) -1 – 2 -2 2 All Real - orNumbers -1.875 Domain 8 -2 -1Range 2(-1) -1 – 2 - 7 y or>– 1.75 4 30, -1.5 ) ( y-intercept 0 20 -1 – 2 - or – 1. 5 2 x-intercept ( 2, -1 0 ) 1 21-1 – 2 Asymptote y 0= -2 2 22-1 - 2 3 23-1 - 2 2 f(x) = 2x-1 - 2 Consider the f(x) = abx-h + k where a = 1 f(x) = x-1 2 -2 Graph the function by transformation using the parent function. h - Horizontal shift k - Vertical shift Parent function f(x) = 2x f(x) = 2x-1 - 2 Consider the f(x) = abx-h + k where a = 1 f(x) = x+4 2 +3 Parent function f(x) = 2x Graph the function by transformation using the parent function. h - Horizontal shift k - Vertical shift f(x) = 2x+4 + 3 Consider the f(x) = abx-h + k where a = 1 f(x) = x-5 2 -6 Parent function f(x) = 2x Graph the function by transformation using the parent function. h - Horizontal shift k - Vertical shift f(x) = 2x-5 - 6 Have an advance reading on the following: What will be the effect to the graph of f(x) = abx-h + k when a > 1 or a < 1? How can we determine the domain and range of the function if we are not to draw the graph? How can we determine the x and y– intercept of the function if we are not to draw the graph? How can we determine the horizontal asymptote of the function if we are not to draw the graph?