Exponential Functions and Graphs Section 5.2 Exponential Functions and Graphs
Objectives Graph exponential equations and exponential functions. Solve applied problems involving exponential functions and their graphs.
Arithmetic vs Geometric Growth Growth is about how a quantity will change over time. The quantity is measured periodically, at even time intervals. For example, at the beginning of each month Arithmetic growth happens when an equal amount is added to a quantity each period Geometric growth happens when a quantity is multiplied by the same amount each period
Simple interest If $100 is deposited in an account which pays 1% simple interest each month then $1 is added to the balance each month $1 is 1% of the initial balance The balance grows like this 100 101 102 103 This is arithmetic growth
Compound Interest If $100 is deposited in a account which earns 1% interest each month but compounded, then the balance grows like this: 100 100 + 1 = 101 101 + 1.01 = 102.01 102.01 + 1.0201 = 103.0301 This isn’t much difference from simple interest at first, but over time the difference will be much larger The balance is multiplied by 1.01 each month, this is geometric growth
Population Growth In the presence of sufficient resources, population tends to grow geometrically The classic example is a population of rabbits where we count only the female rabbits The first year there is one new rabbit A female must be at least one year old to give birth to a new rabbit. But thereafter, that rabbit gives birth to one rabbit per year The count of rabbits is the famous Fibonacci sequence: 1 1 2 3 5 8 13 21 34 … This grows geometrically
Exponential functions Exponential functions model geometric growth So, they are important in many applications Finance Population dynamics
Exponential Function The function f(x) = ax, where x is a real number, a > 0 and a 1, is called the exponential function, base a. We require the base to be positive in order to avoid the imaginary numbers that would occur by taking even roots of negative numbers. The following are examples of exponential functions:
Example Graph the exponential function y = f (x) = 2x. Solution We compute some function values and list the results in a table.
Example (continued) As x increases, y increases without bound. As x decreases, y decreases getting close to 0; as x g ∞, y g 0. The x-axis, or the line y = 0, is a horizontal asymptote. As the x-inputs decrease, the curve gets closer and closer to this line, but does not cross it.
Example Graph the exponential function Note This tells us the graph is the reflection of the graph of y = 2x across the y-axis. Selected points are listed in the table.
Example (continued) As x increases, the function values decrease, getting closer and closer to 0. The x-axis, y = 0, is the horizontal asymptote. As x decreases, the function values increase without bound.
Graphs of Exponential Functions Observe the following graphs of exponential functions and look for patterns in them.
Example a) Graph y = 2x – 2. The graph is the graph of y = 2x shifted to right 2 units.
Example b) Graph f(x) = 2x – 4 The graph of f(x) = 2x – 4 is the graph of y = 2x shifted down 4 units.
Example c) Graph f(x) = 5 − 0.5x The graph is a reflection of the graph of y = 2x across the y-axis, followed by a reflection across the x-axis and then a shift up 5 units.
Application The amount of money A that a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula
Example Suppose that $100,000 is invested at 6.5% interest, compounded semiannually. a. Find a function for the amount to which the investment grows after t years. b. Graph the function. c. Find the amount of money in the account at t = 0, 4, 8, and 10 yr. d. When will the amount of money in the account reach $400,000?
Example (continued) a) Since P = $100,000, r = 6.5%=0.65, and n = 2, we can substitute these values and write the following function
Example (continued) b) Use the graphing calculator with viewing window [0, 30, 0, 500,000].
Example (continued) c) We can compute function values using function notation on the home screen of a graphing calculator.
Example (continued) c) We can also calculate the values directly on a graphing calculator by substituting in the expression for A(t):
Example (continued) Set 100,000(1.0325)2t = 400,000 and solve for t, which we can do on the graphing calculator. Graph the equations y1 = 100,000(1.0325)2t y2 = 400,000 Then use the intersect method to estimate the first coordinate of the point of intersection.
Example (continued) Or graph y1 = 100,000(1.0325)2t – 400,000 and use the Zero method to estimate the zero of the function coordinate of the point of intersection. Regardless of the method, it takes about 21.67 years, or about 21 yr, 8 mo, and 2 days.
The Number e e is a very special number in mathematics. Leonard Euler named this number e. The decimal representation of the number e does not terminate or repeat; it is an irrational number that is a constant; e 2.7182818284…
Example Find each value of ex, to four decimal places, using the ex key on a calculator. a) e3 b) e0.23 c) e0 a) e3 ≈ 20.0855 b) e0.23 ≈ 0.7945 c) e0 = 1
Graphs of Exponential Functions, Base e Example Graph f (x) = ex and g(x) = e–x. Use the calculator and enter y1 = ex and y2 = e–x. Enter numbers for x.
Graphs of Exponential Functions, Base e - Example (continued) The graph of g is a reflection of the graph of f across they-axis.
Example Graph f (x) = ex + 3. The graph f (x) = ex + 3 is a translation of the graph of y = ex left 3 units.
Example Graph f (x) = e–0.5x. The graph f (x) = e–0.5x is a horizontal stretching of the graph of y = ex followed by a reflection across the y-axis.
Example Graph f (x) = 1 e2x. The graph f (x) = 1 e2x is a horizontal shrinking of the graph of y = ex followed by a reflection across the y-axis and then across the x-axis, followed by a translation up 1 unit.