7.1 & 7.2 - Graphing Exponential Transformations Grade Distribution 1st 4th 7th 9th A 15 8 n/a B 10 C 2 3 D F No Shows 100+ 1 Range 78-105 47-101 Avg 89.72 83.5 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

Graphing Exponential Functions Section 7.1 and 7.2, Revised ©2013, vdang@houstonisd.org 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

Identifying Exponential Growth vs Decay Exponential Equation: f(x) = Cax C: COEFFICIENT a: BASE X: EXPONENT Exponential Equations Graph is above the x-axis Domain: (–∞, ∞); Range: (0, ∞) Exponential Growth When the base is greater than 1, a > 1 f(x) is INCREASING Exponential Decay When the base is between zero and one, 0 < a < 1 f(x) is DECREASING 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 1 Determine whether represents exponential growth or decay. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 2 Determine whether represents exponential growth or decay. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn Determine whether represents exponential growth or decay. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

Graphing Exponentials Make a table of values, usually from [–3, 3] Plot the points from the table Connect the dots and draw from left to right, a smooth curve and label any asymptotes 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 3 Graph f(x) = 2x x f(x) –3 (2)–3 1/8 –2 (2)–2 1/4 –1 (2)–1 1/2 (2)0 1 (2)1 2 (2)2 4 3 (2)3 8 x f(x) –3 (2)–3 –2 (2)–2 –1 (2)–1 (2)0 1 (2)1 2 (2)2 3 (2)3 x f(x) –3 –2 –1 1 2 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 3 Graph f(x) = 2x x f(x) –3 (2)–3 1/8 –2 (2)–2 1/4 –1 (2)–1 1/2 (2)0 1 (2)1 2 (2)2 4 3 (2)3 8 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 3 Graph f(x) = 2x (Calculator check) 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 4 Graph f(x) = (1/3)(3)x x f(x) –3 –2 –1 1 2 3 –3 (1/3)(3)–3 1/81 –2 (1/3)(3)–2 1/27 –1 (1/3)(3)–1 1/9 (1/3)(3) 0 1/3 1 (1/3)(3) 1 2 (1/3)(3) 2 3 (1/3)(3) 3 9 –3 (1/3)(3)–2 –2 –1 (1/3)(3)–1 (1/3)(3) 0 1 (1/3)(3) 1 2 (1/3)(3) 2 3 (1/3)(3) 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn Graph f(x) = (1/2)(4)x x f(x) –3 –2 –1 1 2 3 –3 (1/2)(4)–3 1/128 –2 (1/2)(4)–2 1/32 –1 (1/2)(4)–1 1/8 (1/2)(4) 0 ½ 1 (1/2)(4) 1 2 (1/2)(4) 2 8 3 (1/2)(4) 3 32 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 5 Graph f(x) = (1/2)x x f(x) –3 (1/2)–3 8 –2 (1/2)–2 4 –1 (1/2)–1 2 (1/2)0 1 (1/2)1 1/2 (1/2)2 1/4 3 (1/2)3 1/8 x f(x) –3 (1/2)–3 –2 (1/2)–2 –1 (1/2)–1 (1/2)0 1 (1/2)1 2 (1/2)2 3 (1/2)3 x f(x) –3 –2 –1 1 2 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 5 Graph f(x) = (1/2)x x f(x) –3 (1/2)–3 8 –2 (1/2)–2 4 –1 (1/2)–1 2 (1/2)0 1 (1/2)1 1/2 (1/2)2 1/4 3 (1/2)3 1/8 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 6 Graph f(x) = –3(2/5)x (decimals are acceptable) x f(x) –3 –2 –1 1 2 3 –3 –3(2/5)–3 –46.875 –2 –3(2/5)–2 –18.75 –1 –3(2/5)–1 –7.5 –3(2/5)0 1 –3(2/5)1 –1.2 2 –3(2/5)2 -.48 3 –3(2/5)3 -1.92 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn Graph f(x) = 3(2/3)x (decimals are acceptable) x f(x) –3 –2 –1 1 2 3 –3 3(2/3)–3 10.125 –2 3(2/3)–2 6.75 –1 3(2/3)–1 4.5 3(2/3)0 3 1 3(2/3)1 2 3(2/3)2 5/3 3(2/3)3 .889 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Translations Vertical Translations Moving the graph up and down; the change is OUTSIDE the exponent Moving UP is when the translated is added, Moving DOWN is when the graph is subtracted Horizontal Translations Moving the graph up and down; the change is WITHIN the exponent Moving LEFT is when the translated is added, Moving RIGHT is when the graph is subtracted 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 7 If f(x)=2x, determine the transformation to g(x) = 2x + 2. State the equation for the asymptote, domain and range in Interval Notation. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 8 If f(x)=2x, determine the transformation to g(x) = 2x + 2. State the equation for the asymptote, domain and range in Interval Notation. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn If f(x)=2x, determine the transformation g(x) = 2x + 3 – 4. State the equation for the asymptote, domain and range in Interval Notation. 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

Vertical Stretches and Compressions Applied to the coefficient of y = Cax Stretches when |c| > 1 Compresses when |c| < 1 Horizontal Stretches and Compressions Applied to the number in front of the exponent, d, of y = Cadx Take the horizontal stretch/compression and take the reciprocal Stretches when 0 < d < 1 Compresses when d > 1 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 9 If f(x) = 2x, determine the transformation to g(x) = (1/4)(2)x 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 10 If f(x) = 2x, determine the transformation to g(x) = (4)(2)x 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn If f(x) = 2x, determine the transformation to g(x) = (–5)(2)x 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 11 If f(x) = 2x, determine the transformation to g(x) = (2)0.25x 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn If f(x) = 2x, determine the transformation to g(x) = (2)4x 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 12 If f(x) = 2x, determine the transformation g(x) = (4) (2)x – 1 – 3. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x f(x) –2 –1 1 2 3 x f(x) –3 (4) (2)–3 – 1 – 3 –2.75 –2 (4) (2) –2 – 1 – 3 –2.5 –1 (4) (2)–1 – 1 – 3 (4) (2) 0 – 1 – 3 1 (4) (2)1 – 1 – 3 2 (4) (2)2 – 1 – 3 5 3 (4) (2)3 – 1 – 3 13 x f(x) –3 (4) (2)–3 – 1 – 3 –2 (4) (2) –2 – 1 – 3 –1 (4) (2)–1 – 1 – 3 (4) (2) 0 – 1 – 3 1 (4) (2)1 – 1 – 3 2 (4) (2)2 – 1 – 3 (4) (2)x – 1 – 3 3 (4) (2)3 – 1 – 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 12 If f(x) = 2x, determine the transformation g(x) = (4) (2)x – 1 – 3. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x f(x) –3 (4) (2)–3 – 1 – 3 –2.75 –2 (4) (2) –2 – 1 – 3 –2.5 –1 (4) (2)–1 – 1 – 3 (4) (2) 0 – 1 – 3 1 (4) (2)1 – 1 – 3 2 (4) (2)2 – 1 – 3 5 3 (4) (2)3 – 1 – 3 13 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 13 If f(x) = 2x, determine the transformation g(x) = –(1/2)x + 1 – 2. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x f(x) –3 –(1/2)–3 + 1 – 2 –18 –2 –(1/2)–2 + 1 – 2 –10 –1 –(1/2)–1 + 1 – 2 –6 –(1/2)0 + 1 – 2 –4 1 –(1/2)1 + 1 – 2 2 –(1/2)2 + 1 – 2 –2.5 3 –(1/2)3 + 1 – 2 –2.25 x f(x) –3 –2 –1 1 2 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn If f(x) = 2x, determine the transformation g(x) = 2(2/3)x – 1 + 3. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x f(x) –3 2(2/3)–3 – 1 + 3 13.125 –2 2(2/3)–2 – 1 + 3 9.75 –1 2(2/3)–1 + 1 + 3 7.5 2(2/3)0 + 1 + 3 6 1 2(2/3)1 + 1 + 3 5 2 2(2/3)2 + 1 + 3 4.33 3 2(2/3)3 + 1 + 3 3.889 x f(x) –3 –2 –1 1 2 3 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

Growth/Decay Factor Equation Equation: A = P (1 + r)t P: Initial Principal +: Growth Factor -: Decay Factor r: Interest Rate t: Time it takes to accrue amount 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 14 You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years? A = ? Do you know much you are going to make? P = $5,000 $5,000 is deposited r = 0.10 Interest Rate – remember it needs to be in decimal form t = 3 Time it takes to accrue amount + Growth 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 14 You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years? 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 15 You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years? A = ? Do you know much you are going to make? P = $5,000 $5,000 is spent r = 0.10 Interest Rate – remember it needs to be in decimal form t = 3 Time it takes to accrue amount Decay Decreasing Value 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Example 15 You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years? 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Your Turn You buy a car that cost $10,000 and depreciates 10% per year. How much money have you lost, compared to the original amount, in 5 years? 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations

7.1 & 7.2 - Graphing Exponential Transformations Assignment Pg 482: 3-5 all, Graph, Label domain and range, and asymptotes for 7-23 EOO, 24, 25, 27, 28, 30 Pg 489: 3-6 all, Graph, Label domain and range, and asymptotes for 7-23 EOO, 26, 27, 31a, 31c 12/26/2018 1:37 PM 7.1 & 7.2 - Graphing Exponential Transformations