Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is called a. The set of all possible inputs of a function is called the. The set of all possible outputs of a function is called the. Eample. Determine which equations specify functions with independent variable. For those equations that define functions, determine their domains. For those equations that do not define functions, specify an -value such that it corresponds to more than one y-value. (a) + 5 y = 10 (b) y = (c) y y = 1 Eample. Which graph represents a function? (a) (b) (c) Vertical Line Test: Section -1: Functions
Function Notation The notation f () represents the output of the function that corresponds to the input. f is the name of the function. f () is often epressed as the single variable. Eample. For f ( ) =, find (a) f (1) (b) f ( 1 + h) (c) f ( 1 + h) f (1) h (d) f ( + h) f ( ) h Eample. Translate the verbal definition of a function into an algebraic function: Function h multiples the square of the domain element by 0 and subtracts five times the domain element from the result. Eample. Translate the algebraic definition of a function into a verbal definition: 3 + f ( ) = Section -1: Functions
Section - Elementary Functions: Graphs and Transformations Goal: To review graphs of elementary functions, graph transformations of these functions, and graph piecewise functions Eample. Graph the following elementary functions on the aes provided. (a) y = (b) y = (c) 3 y = (d) y = (e) y = 3 (f) y = Graphing Transformations of y = f () Assume h > 0, k > 0, and A > 1. shifts the graph of y = f () right h units. shifts the graph of y = f () left h units. shifts the graph of y = f () up h units. shifts the graph of y = f () down h units. reflects the graph of y = f () around the -ais. epands the graph of y = f () vertically by multiplying each ordinate value by A. contracts the graph of y = f () vertically by multiplying each ordinate value by A. Section -: Elementary Functions: Graphs and Transformations
Eample. Indicate verbally how the graph of each function related to the graph of one of the si basic functions. Sketch a graph of each function. (a) g ( ) = 3 + 5 (b) h ( ) = + 4 Eample. Use the graph of y = k() shown below to graph the following functions. Specify the coordinates of the three points that correspond to three points shown on the graph below. (a) k ( + 1) 5 1 (b) k ( ) + 3 (c) k ( 4) Definition: A function is a function with different rules for different parts of its domain. 3, 0 Eample. Graph the function f ( ) =, + 3, < 0 drawing open and closed circles where appropriate. Eample. A store developed a promotion to attract shoppers. If patrons spend between 0 and $10, then they receive 5% off the price. If patrons spend more than $10, but less than $30, then they receive 10% off. If patrons spend $30 or more, then they receive 0% off. Define a piecewise function g () that represents the discount (in dollars) when dollars are spent. Section -: Elementary Functions: Graphs and Transformations
Section -3 Quadratic Functions Goal: To graph quadratic equations, identify the intercepts and maimum and minimum values of quadratic functions, solve quadratic equations and inequalities Quadratic Function A quadratic function is any function that can be written in the form (standard form), where a, b, and c are real numbers with a 0. The standard form of a quadratic formula can be changed to verte form by completing the square: (verte form). The graph of any quadratic function is a. The maimum/minimum value of a parabola is located at its verte. The verte of the graph of a quadratic in standard form is located at the point (verte formula). The verte of the graph of a quadratic in verte form is located at the point. Eample. Complete the square to determine the verte form of the given quadratic function. Use the resulting form to determine the verte of the parabola. Check your answer by using the verte formula. (a) f ( ) = + 6 3 (b) g ( ) = 1 + 6 Quadratic Formula To solve a quadratic equation of the form a + b + c = 0, a 0, you can factor the quadratic equation and use the zero product property to solve the resulting linear equations or you can use the quadratic formula: =, provided that. Eample. Find the intercepts, verte, maimum/minimum value, and the range of the following quadratic functions. Sketch the graph of each function. (a) y = ( + 3) 5 (b) ( 1) y = + 4 (c) y = + 10 10 Section -3: Quadratic Functions
Eample. For the functions f ( ) = 0 and g ( ) = + 5, (a) Graph f and g in the same coordinate system. (b) Solve f ( ) = g( ) algebraically to two decimal places. (c) Solve f ( ) > g( ) using parts (a) and (b). Give your answer in interval notation. (d) Solve f ( ) < g( ) using parts (a) and (b). Give your answer in interval notation. Eample. Suppose the weekly profit generated from the sale of copies of a particular brand of video editing software is modeled by the equation P ( ) = + 550 000, where 0 510. What is the maimum weekly profit that can be attained? Section -3: Quadratic Functions
Section -4 Eponential Functions Goal: To graph basic eponential functions, solve eponential equations with the same base, and to solve application involving compound interest Definition: Eponential Function The equation f ( ) = b, where b > 0 and b 1, defines an eponential function for each different constant b called the. The domain of f is and the range of f is. Theorem 1: Basic Properties of the Graph of f ( ) = b, b > 0, b 1 1. All graphs will pass through the point.. All graphs are continuous on the interval. 3. The horizontal asymptote is. 4. f() is an increasing function if. 5. f() is a decreasing function if. Sketch the graph of b y = for b > 1. Sketch the graph of b y = for 0 < b < 1. Eample. Use transformations to graph each eponential function below. (a) y = 4 1 (b) y = 3 Section -4: Eponential Functions
Theorem : Properties of Eponential Functions For positive numbers a and b, where a 1, b 1 and real numbers and y, 1. a y a a = = y a a ( ab) = = b ( a ) = y. y a = a if and only if =. 3. For 0, a = b if and only if =. Eample. Solve each equation for. (a) 3 = 3 7 1 (b) 6 1 = 36 (c) e e = 0 Interest Formulas Simple interest means you only receive interest on your initial investment (i.e., the principal). Compound interest means you receive interest on your interest. If A = amount in account after t years, r = annual interest rate (epressed as a decimal), m = number of times account is compounded in a year, and P = principal (i.e., initial investment), then the amount in the following accounts after t years: Simple interest: A = Compound interest: A = Ideally, we want m to be as large as possible. If m grows very large, we come up with the following formula Continuous compounding: A = Eample. Jay wants to invest $500 for 5 years. Should he invest money in an account compounded quarterly at an annual rate of 3.75% or should he invest his money in an account compounded continuously at an annual rate of 3.5%? Eample. How much should Mary and Joey invest now in an account compounded semiannually at an interest rate of 4.5% in order to end up with $50,000 in 15 years? Section -4: Eponential Functions
Section -5 Logarithmic Functions Goal: To graph basic logarithmic functions, use log properties, and solve logarithmic and eponential equations Definition: The inverse of an eponential function is a function. For b > 0 and b 1, y = logb is equivalent to the epression. In words, log represents. b The domain of y = logb is and its range is. Eample. Rewrite the following equations in an equivalent logarithmic form. (a) 5 = 5 (b) r D = C Eample. Use the logarithm definition in words to evaluate the following epressions without a calculator. (a) log 4 64 (b) log 5 5 Theorem 1: Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and are real numbers, then 1. log b 1 = 5. log b MN =. log b b = M 6. log 3. log b b = b = N b 4. b log p 7. = = log b M 8. log M = log N if and only if b b Eample. Use log properties to epand the following epressions as much as possible. (a) ln y z (b) log b v Section -5: Logarithmic Functions
Eample. Solve each equation for. (a) logb = logb 5 + 3logb 3 (b) log( + 3) log = 1 We can solve eponential equations by taking the log of both sides of the equation, and then using log properties to simplify the epressions. Eample. Solve each equation for. (a) 5 = 0 (b) e = 4 Eample. How long will it take money invested in an account compounded continuously at an interest rate of 5.6% to double? Eample. Shuan has saved $000. At what interest rate must her money be invested in order for her money to grow to $3000 in 1 year if the account is compounded continuously? Section -5: Logarithmic Functions