College Algebra MATH 123
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This 10 page Class Notes was uploaded by Lisa Wisoky on Monday October 12, 2015. The Class Notes belongs to MATH 123 at Fayetteville State University taught by Wu Jing in Fall. Since its upload, it has received 71 views. For similar materials see /class/221583/math-123-fayetteville-state-university in Mathematics (M) at Fayetteville State University.
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Date Created: 10/12/15
Chapter 4 Linear and Quadratic Functions 41 Linear Functions and Their properties De nition A linear function is a function of the form f x mx b The graph of a linear function is a line with slope m and y intercept b Its domain is the set of all real numbers Example 1 Graph the linear function f x 3x 7 Average rate of change Linear functions have a constant average rate of change That is the average rate of change A ofa linear lnctlon fx mx b is E m Example 2 Find the average rate of change of f x 3x 7 Increasing decreasing and constant linear functions Let f x mx b be a linear function It is increasing over its domain if its slope is positive ie mgt0 It is decreasing over its domain if its slope is negative ie mlt0 It is constant over its domain if its slope is zero ie m0 Example 3 Determine whether the following linear functions are increasing decreasing or constant 1 fx5x 2 2 gx 3x8 3 st 3 Zt 4 4 hz7 Example 4 Suppose that the quantity supplied S and quantity demanded D of cell phones each month are given by the following functions Sp 60p 900 Dp 15p2850 where p is the price in dollars of the phone 1 The equilibrium price of a product is defined as the price at which quantity supplied equals quantity demanded That is the equilibrium price is the price at which S p D p Find the equilibrium price of cell phone What is the equilibrium quantity the amount demanded or supplied at the equilibrium price 2 Determine the prices for which quantity supplied is greater than quantity demanded That is sole the inequality Sp gt Dp 3 Graph S SpD Dp and label the equilibrium price 42 Building Linear Functions from Data Draw and interpret scatter diagrams Example 1 The data listed below represent the apparent temperature versus the relative humidity in a room whose actual temperature is 720 1 Draw a scatter diagram by hand treating relative humidity as the independent variable 2 Describe what happens to the apparent temperature as the relative humidity increase Distinguish between linear and nonlinear relations Example 2 Determine whether the relation between the two variables below is linear or nonlinear Finding an equation for linearly related date Example 3 Using the table in Example 1 1 Select two points and nd an equation of the line containing the points 2 Graph the line on the scatter diagram obtained in Example 11 3 Find the line of best t using a graphing utility 4 Graph the line of best t on the scatter diagram obtained in Example 11 43Quadratic Functions and Their Properties De nition A quadratic function is a function of the form fx ax2 bxc 61 0 Its domain is the set of all real numbers Properties of the Graph of a Quadratic Function f x ax2 bx c a 7i 0 Vertex if i Axis ofsymmetry line x i 2a 2a 2a Parabola opens up if a gt 0 the vertex is a minimum point Parabola opens down if a lt 0 the vertex is a maximum point Example 1 Without graphing locate the vertex and axis of symmetry of the parabola de ned by fx 3x2 6x 1 Does it open up or down Example 2 Use the information from Example 1 and the locations of the intercepts to graph fx 3x2 6xl Example 3 1 Graph f x x2 6x 9 by determine whether the graph opens up or down Find its vertex axis of symmetry yintercept and xintercepts if any 2 Determine the domain and the range of f 3 Determine where f is increasing and where it is decreasing Example 4 1 Graph f x 2x2 x 1 by determine whether the graph opens up or down Find its vertex axis of symmetry yintercept and xintercepts if any 2 Determine the domain and the range of f 3 Determine where f is increasing and where it is decreasing Find the maximum 01 minimum value of a quadratic function Forfx ax2 bxc 61 0 o If a lt 0 then f is the maximum value of f a o If a gt 0 then f is the minimum value of f a Example 5 Determine whether the quadratic function f x x2 4x 7 has a maximum or minimum value Then nd the maximum or minimum value 44 Quadratic Models Building Quadratic Functions from Data Example 1 The marketing department at Texas Instruments has found that when certain calculators are sold at a price of p dollars per unit the number x of calculators sold is given by the demand equation x 21000 150p a Express the revenue R as a function of the price p b What unit price should be established in order to maximize revenue 7000 c If this price is charged what is the maximum revenue 735000 d How many units are sold at this price 10500 e Graph R Example 2 A farmer has 2000 yards of fence to enclose a rectangular eld What are the dimensions of the rectangle that encloses the most area 500 by 500 Example 3 A projectile red from a cliff 500 feet above the water at an inclination of 450 to the horizontal with a muzzle velocity of 400 feet per second In physics it is established that the height h of the projectile above the water is given by Z hx 32x2 x 500 400 where x is the horizontal distance of the projectile from the base of the cliff 1 Find the maximum height of the projectile l750ft 2 How far away from the base of the cliff will the projectile strike the water 5458ft 45 Inequalities Involving Quadratic Functions Solve inequalities involving a quadratic function 1 To solve the inequality ax2 bx c gt 0 a 7i 0 we graph the function f x ax2 bx c and from the graph determine where it is above xaXis that is where f x gt 0 2 To solve the inequality ax2 bx c lt 0 a 7i 0 we graph the function f x ax2 bx c and from the graph determine where it is below xaXis that is where fx lt 0 3 If the inequality is not strict we include the Xintercepts in the solution Example 1 Solve the inequality x2 S 4x 12 and graph the solution set Example 2 Solve the inequality 2x2 S x 10 and graph the solution set
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