3.1 Exponential Functions and Their Graphs Date: Exponential Function

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1 3.1 Exponential Functions and Their Graphs Date: Exponential Function Exponential Function: A function of the form f(x) = b x, where the b is a positive constant other than, and the exponent, x, is a variable. Notice: there is no single parent exponential function because each choice of the base b determines a different function. Evaluating Exponential Functions Learning Target A: I can evaluate exponential functions given an input value. Example 1. Evaluate each exponential function with the given input value(s). A) f(x) = 4 x ; where x = 3 B) f(x) = 2 x ; where x = 3 C) f(x) = 2 3 x ; where x = 1 D) f(x) = x ; where x = 2 Simplifying Exponential Expressions with Base e Learning Target B: I can simplify exponential expressions with base e. Example 2. Simplify each expression. A) e 3 e 5 B) (2e 5x ) 3 C) 18e5 27e 2 D) ( 48e2 12e 4) 2 E) 25e 8x F) ( 4e 3 ) 4 1

2 Solving Exponential Equations with Common Bases on Each Side Learning Target C: I can solve exponential equations with common bases on each side. One method we can use to solve exponential equations is to rewrite the equation with common bases on both sides. For example: 9 x = 3 x+1 both sides of the equation have x in the exponent (3 2 ) x = 3 x+1 since 9 = 3 2, we can rewrite the left side of the equation to have a base of 3 like the right side 3 2x = 3 x+1 therefore, we have the equation 2x = x + 1 using the Power Property of Exponents, we can simplify the left side to solve Example 3. Solve each exponential equation by rewriting it with common bases on both sides. A) 2 x = 8 B) 3 x = 81 C) 4 2x = 16 D) ( 1 2 )x = 8 E) 64 = 4 2x 1 F) ( 1 3 ) x = 27 G) 27 x+2 = 1 81 H) ( 3 4 )x =

3 Graphing Exponential Functions Learning Target D: I can sketch the graph of an exponential function by hand. Exponential Functions can be increasing, exponential growth, or decreasing, exponential decay. f(x) = b x when b > 1; and when 0 < b < 1; A) Graph each function using the table of values. x f(x) = 2 x x f(x) = ( 1 2 ) x Domain: 1 Domain: 2 Range: 2 Range: 3 3 B) Give the end behavior for each function. What do they have in common? C) Identify the asymptote on each graph. D) What is the relationship between the two graphs? E) How else could you write the function f(x) = ( 1 2 )x? F) Why is the y-intercept the same on both graphs? 3

4 General Equation for a Transformed Exponential Function: g(x) = a(b) x h + k Finding the Horizontal Asymptote: Because the horizontal asymptote is affected by the translation, its equation is always:. Example 4. Given each exponential function, identify its base and whether it is exponential growth or decay. Then sketch its graph by finding its asymptote and 2-3 points on the curve. A) f(x) = 3 x B) f(x) = (2) x C) f(x) = 4 x D) f(x) = 3 ( 1 2 )x E) f(x) = e x 2 F) f(x) = 2 ( 1 4 )x Writing Exponential Functions from Their Graphs Learning Target E: I can write a simple exponential function of the form f(x) = b x given its graph. Example 5. Given each graph, the value of b to write an exponential function of the form f(x) = b x. A) B) 4

5 Exponential Growth and Decay in the Real World Learning Target F: I can use exponential functions to model real world exponential growth and decay situations. Exponential Growth Function: f(t) = a(1 + r) t, where a > 0 and r is a constant percent increase (expressed as a decimal) for each unit increase in time t. Growth Factor: the base 1 + r Growth Rate: constant percent increase, r, in decimal form Example 6. Tony purchased a rare guitar in 2000 for $12,000. Experts estimated that its value would increase by 14% each year. A) Write a function to model the situation. B) How much was the guitar worth in 2003? C) Use a graph to find the number of years it would take for the value of the guitar to reach $60,000. Exponential Decay Function: f(t) = a(1 r) t, where a > 0 and r is a constant percent decrease (expressed as a decimal) for each unit increase in time t. Decay Factor: the base 1 r Decay Rate: constant percent decrease, r, in decimal form Example 7. The value of a truck purchased new for $28,000 decreases by 9.5% each year. A) Write a function to model the situation. B) How much was the truck worth 5 years after it was purchased? C) Predict after how many years the value of the truck will be $

6 Compound Interest Learning Target G: I can model and solve compound interest situations with exponential functions. Compounding Interest n Times per Year V(t) = P (1 + r n ) nt V(t) = the value V of the investment at time t P = principal (the amount invested) r = annual interest rate (as a decimal) n = number of times interested is compounded in one year Compounding Interest Continuously V(t) = Pe rt V(t) = the value V of the investment at time t P = principal (the amount invested) r = annual interest rate (as a decimal) Example 8. Write a function to model and solve each situation. A) A person invests $3500 in an account that earns 3% annual interest. Function: i) What is the value of the investment after 5 years? ii) After how many years will the investment reach $10,000? B) A person invests $1200 in an account that earns 2% annual interest compounded quarterly. Function: i) What is the value of the investment after 20 years? ii) After how many years will the investment reach $1500? C) A person invests $600 in an account that earns 6.25% annual interest compounded semi-annually. i) Find when the value of the investment doubles. Function: D) A person invests $5000 in an account that earns 3.5% annual interest compounded continuously. Function: i) What is the value of the investment after 6 years? ii) Find when the value of the investment reaches $12,000. 6

2.4 - Exponential Functions

c Kathryn Bollinger, January 21, 2010 1 2.4 - Exponential Functions General Exponential Functions Def: A general exponential function has the form f(x) = a b x where a is a real number constant with a