Nonlinear Functions Chapter 10

Nonlinear Functions 10.2 Quadratic Functions 10.4 Exponential Functions 10.5 Logarithmic Functions

10.2 Quadratic Functions A linear function is defined by f(x) = ax + b A quadratic function is defined by f(x) = ax 2 + bx + c where a, b, and c, are real numbers, with a  0, In a quadratic function, the independent variable is squared. Simplest quadratic function is f(x) = x 2, with a = 1, b = 0, and c = 0.

Quadratic Functions The graph of a quadratic function is a parabola.

Quadratic Functions f(x) = x 2 Vertex

Quadratic Functions The graph of a quadratic function is a parabola.

Quadratic Functions h(x) = -.2x 2 Vertex

Quadratic Functions The graph of a quadratic function f(x) = ax 2 + bx + c is a parabola with vertex ( h, k ), where If a > 0, the parabola opens upward. Vertex is its lowest point If a < 0, the parabola opens downward. Vertex is its highest point The graph of a quadratic function f(x) = ax 2 + bx + c is a parabola with vertex ( h, k ), where If a > 0, the parabola opens upward. Vertex is its lowest point If a < 0, the parabola opens downward. Vertex is its highest point

Example Determine if the graph of the following quadratic function opens downward or upward, and identify the vertex (h, k ). f(x) = 2 x 2 – 7 x + 12 a = 2, b = -7 ( a > 0,  graph opens upward) ( h, k ) = (7/4, 47/8)

Example ( h, k ) = (7/4, 47/8)  f(x) = 2 x 2 – 7 x + 12

Now You Try According to data from the National Highway Traffic Safety Administration, the fatal accident rate as a function of the age of the driver in x years is approximated by the function for 16 ≤ x ≤ 88. At what age is the fatal accident rate the lowest?

Applications of Quadratic Functions The vertex of a parabola y = ax 2 + bx + c can be used in applications involving maximum or minimum values. Profit Cost Time

Examples The revenue of a charter bus company depends on the number of unsold seats. If the revenue R ( x ) is given by R ( x ) = x – x 2 where x is the number of unsold seats, find the maximum revenue and the number of unsold seats that corresponds to maximum revenue.

Examples R ( x ) = x – x 2 R ( x ) = – x x a = - 1, b = 50 Maximum revenue of $ occurs when there are 25 unsold seats.

Examples R ( x ) = – x x (25, 5625)

Examples The supply function for a commodity is given by p = q , and the demand function is given by p = -10 q What is the equilibrium quantity and the equilibrium price?

Examples At equilibrium, q = -10 q Write the formula in standard quadratic form q q = 0, and solve using the quadratic formula with a = 1, b = 10, and c = -3000

Examples Equilibrium point = (50, 2700) Equilibrium quantity = 50 Equilibrium price = 2700

Examples (50, 2700) p = q p = -10 q

Now You Try Shannise Cole makes and sells candy. She has found that the cost per box for making x boxes of candy is given by a. How much does it cost per box to make 15 boxes? 18 boxes? 30 boxes? b. How many boxes should she make in order to keep the cost per box at a minimum? What is the minimum cost per box?

Try Another Find the equilibrium quantity and equilibrium price for the commodity whose supply and demand functions are: Supply: p = q 2 + q + 10 Demand: p = - 10 q

an exponential function is a rule in which a constant is raised to variable power; f(x) = a x, where a > 0 and a  1 Example: You start a company in a dynamic new industry and discover your sales double each year. Describe this with a function. Sales = 2 x, where x = year Note: a linear function will not reflect the doubling each year Exponential Functions

Sales = 2 x

(a, b > 0) Properties of Exponents (a, b > 0)

Properties of Exponents If a > 0, a  1, and a x = a y, then x = y. Also, if x = y, then a x = a y Solve 9 x = 27 Solution First rewrite both sides of the equation so the bases are the same. Since 9 = 3 2 and 27 = 3 3, (3 2 ) x = x = 3 3 2x = 3 x = 3/2 Therefore, 9 3/2 = 27

Properties of Exponents Solution: 32 = 2 5, 128 = 2 7 (2 5 ) 2x – 1 = (2 7 ) x x – 5 = 2 7x x – 5 = 7x x = 26 x = 26/3 Now You Try. Solve for x: 32 2x – 1 = 128 x + 3

Graphing an Exponential Function an exponential function, y = a x, graphs as follows if a > 1: function rises from left to right - exponential growth

Graphing an Exponential Function x2x2x

Graphing an Exponential Function an exponential function, y = a x, graphs as follows If a > 1: function rises from left to right - exponential growth The larger the base a, the more steeply the graph rises to the right.

Exponential Growth, a > 1 y = 3 x y = 2 x y = 3/2 x 1.The graph is above the x- axis. 2.The y- intercept is 1. 3.The graph climbs steeply to the right. 4.The negative x- axis is a horizontal asymptote. 5.The larger the base a, the more steeply the graph rises.

Graphing an Exponential Function an exponential function, y = a x, graphs as follows If a > 1: function rises from left to right - exponential growth, If 0 < a < 1; function falls from left to right - exponential decay

Graphing an Exponential Function

an exponential function, y = a x, graphs as follows If a > 1: function rises from left to right - exponential growth, If 0 < a < 1; function falls from left to right - exponential decay The smaller the base a, the more steeply the graph falls to the right.

Exponential Decay, 0 < a < 1 1.The graph is above the x- axis. 2.The y- intercept is 1. 3.The graph falls sharply to the right. 4.The positive x- axis is a horizontal asymptote. y = ( 1/2) x y = ( 1/4) x y = ( 1/8) x 5.The smaller the base a, the more steeply the graph falls.

Graphing an Exponential Function Horizontal reflection of the graph of f(x) = 2 x

Graphing an Exponential Function Vertical reflection of the graph of f(x) = 2 x

Graphing an Exponential Function Graph of f(x) = -2 x translated vertically 3 units

10.5 Logarithmic Functions LOGARITHM For a > 0, a  1, and x > 0, y = log a x means a y = x y = log a x is read as “y is the logarithm of x to the base a.” Ask yourself, “To what power do I raise a to get x?”

Equivalent Expressions Exponential FormLogarithmic Form (a) 2 3 = 8Log 2 8 = 3 (b) (1/5) -2 = 25log 1/5 25 = -2 (c) 10 3 = 1000log = 3 (d) 4 -3 = 1/64log 4 1/64 = -3 (e) e 0 = 1log e 1 = 0

Properties of Logarithms (Product property) (Quotient property) (Power property)

Properties of Logarithms

Logarithms The two most common types of logarithms 1. Logarithms to base 10: called common logarithms log 10 x is abbreviated log x 2. Logarithms to base e: called natural logarithms log e x is abbreviated ln x

Logarithms Change of Base Theorem For any positive numbers a and x (with a  1),

Logarithmic and Exponential Equations Worldwide production of automobiles has been steadily increasing. The number of people per automobile is approximated by f ( x ) = 24.2(.9792 x ), where x = 0 corresponds to According to this model, when will the number of people per car reach 7? To find x such that f ( x ) = 7, solve: 24.2(.9792 x ) = 7

Solving Exponential Equations For x > 0, y > 0, and b  1, if x = y, then log b x = log b y and if log b x = log b y, then x = y

Solving Exponential Equations The number of people per automobile will reach 7 in 2019 ( ) (Power property) (if x = y, then log b x = log b y)

Now You Try According to projections by the U.S. Census Bureau, the world population (in billions) in year x is approximated by the function In what year will the population reach 7 billion?

Chapter 10 End 