Solutions of Equations An ordered pair will be a solution to an equation if the equation is when the numbers are substituted into the equation.

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1 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 1: Graphs, Functions, and Models Section 1.1 Introduction to Graphing Solutions of Equations An ordered pair will be a solution to an equation if the equation is when the numbers are substituted into the equation. Example 1 Use substitution to determine whether the ordered pair (3.75, -3.25) is a solution of the equation x 2 + y 2 = 5. Graphing & Intercepts An x-intercept is a point at which the graph crosses the x-axis in the form (a, 0). To find a, let y = 0 and solve for x. A y-intercept is a point at which the graph crosses the y-axis in the form (0, b). To find b, let and solve for y. Example 2 Use the intercepts to graph the equation 3x 7y = 21 Example 3 Graph the equation y = ½ x + 2 Example 4 Graph the equation y = -x using the TABLE feature of a graphing calculator. Fill in the y-values for x = -3, -2, and -1. Circles The equation of a circle with center (h, k) and radius r, in standard form, is Example 5 circle. Find the center and radius of the circle (x 4) 2 + y 2 = 7 and then graph the Example 6 Find an equation for a circle satisfying the given conditions: center (0, 0) passing through (-4, 6)

2 Section 1.2 Functions and Graphs 2 Function A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly member of the range. Graphically speaking, the domain is the set of all x-coordinates and the range is the set of all y-coordinates. Example 1 Example 2 Grocery Store examples Mother/Offspring example Example 3 Determine whether the relation is a function. Identify the domain and range. {(5, 0), (3, -1), (0, 0), (-5, -1), (4, -2)} Function Notation f(x) is read f of x, or f at x, or the value of f at x and is another name for y. Example 4 A graph of a function is shown. Using the graph, find the indicated function values. a) f(4) b) f(0) y = f(x) c) f(-4) Example 5 Let f(x) = x 2 3x + 4 and g(x) = 3x + 1. Find the following. a) f(2) b) g(a) c) f(a + 2)

3 Vertical Line Test If it is possible for a vertical line to cross or touch a graph more than once, then the graph is the graph of a function. 3 Example 6 Do the following graphs represent functions? Why or why not? a) b) Finding Domains of Functions The domain of a function is the set of all numbers that can be plugged in for x to arrive at an answer for y. Watch out for dividing by zero within fractions and taking even roots of negative numbers. Example 7 Graph the following functions and find the domain and range of each. Use both interval notation and set notation. a) f(x) = x + 5 b) c) d) f(x) = (x 2) e)

4 Section 1.3 Linear Functions, Slope, and Applications 4 Slope of a Line Slope is a numerical measurement of the steepness of a line. The slope m of a line containing the points (x 1, y 1 ) and (x 2, y 2 ) is given by rise thechangeiny y m run thechangeinx x Example 1 Find the slope of the line containing the given points. a) (-6, -1) and (2, -13) b) 2 2 y x 1 1 c) (3, -5) and (3, -9) Slope-Intercept Equation of a Line: y = mx + b The constant m is the and the y-intercept is (0, b). Example 2 Graph the linear equation and determine its slope, if it exists. a) 2y x = 8 b) x = 3 c) y = ¾

5 Horizontal and Vertical Lines Horizontal lines are given by equations of the type y = b. They are functions. 5 Vertical lines are given by equations of the type x = a. They are not functions. x = 2 y = 2 Types of Slope Positive line slants up from left to right Negative line slants down from left to right Zero horizontal line Undefined vertical line Linear Functions A function f is a linear function if it can be written as f(x) = mx + b. A function is linear if its variables are raised to the power. Example 3 Superior Cable Television charges a $65 installation fee and $80 per month for deluxe service. Write an equation that can be used to determine the total cost, C(t), for t months of deluxe cable service. Then find the total cost for 8 months of service. Average Rate of Change Slope gives the average rate of change in y per unit change in x, where the value of y depends on the value of x. Example 4 The percent of 10 th -grade students who have smoked daily in the last 30 days has greatly decreased, from 16.3% in 1995 to 8.3% in 2004 (Source: Monitoring the Future, University of Michigan Institute for Social Research and National Institute on Drug Abuse). Find the average rate of change over the 9-yr period in the percent of 10 th -grade students who have smoked daily in the last 30 days. Round to the nearest hundredth.

6 Section 1.4 Equations of Lines and Modeling 6 Slope-Intercept Equation of a Line: y = mx + b The constant m is the slope and the y-intercept is (0, b). Example 1 Find the slope and y-intercept of the graph of the linear equation. Then write the equation of the line in slope-intercept form. Point-Slope Equation of a Line: The constant m is the slope and (x 1, y 1 ) is point the line passes through. Example 2 Write a slope-intercept equation for a line with slope of 2/3, passing through (-4, -5). Example 3 Write a slope-intercept equation for the line passing through the points (2, 7) and (-1, -8). Parallel and Perpendicular Lines Nonvertical lines are parallel if and only if they have the slope and different y-intercepts. Two lines, neither of which is vertical, are perpendicular if and only if their slopes have a product of -1. Their slopes are called opposite or negative reciprocals. Example 4 Find the equation in slope-intercept form of the line that passes through the point (2, 4) and satisfies the following conditions: a) parallel to the line 3x + 4y = 12 b) perpendicular to the line 3x + 4y = 12

7 Modeling Example 5 America. The table below illustrates the upward trend to choose cremation in 7 a) Model the data with a linear function. Let the independent variable represent the number of years after Use the data points (0, 32.2) and (3, 35.3). Year, x Percentage of Deaths Followed by Cremation, y 2005, % 2006, Projected Source: Cremation Association of North America 2007, , , b) Using the function found in part (a), estimate the percentage of deaths followed by cremation in Regression Lines We will be finding the line of best fit called the regression line or regression equation using the graphing calculator. The coefficient of correlation, r, describes the strength of the linear relationship between x and y. The closer is to 1, the better the correlation. A positive value of r indicates that the regression line has a positive slope, and a negative value of r indicates that the regression line has a negative slope.

8 TI Graphing Calculator Keystrokes for Curve Fitting Turn on correlation coefficient (this just needs to be done once): 2 nd 0 scroll to DiagnosticON ENTER ENTER (the calculator will say Done) 8 Clear lists: STAT 1:Edit put cursor on L1 CLEAR cursor down put cursor on L2 CLEAR cursor down Enter data: enter x-values in L 1 and then y-values in L 2 (make sure that there are an equal number of entries in L 1 and L 2 ) Finding equation of best fit: STAT cursor right to CALC LinReg (ax + b) ENTER ENTER (the screen should say LinReg at the top and then the equation y = ax + b and then values for a & b & r) In general, the basic steps are: STAT EDIT STAT CALC Example 6 America. The table below illustrates the upward trend to choose cremation in a) Use a graphing calculator to fit a regression line to the data. Let the independent variable represent the number of years after Write the equation of the regression line, rounding decimals to three places. Year, x Percentage of Deaths Followed by Cremation, y 2005, % 1 Projected Source: Cremation Association of North America 2006, , , , b) Find the coefficient of correlation, rounding to the nearest thousandth. Is the regression line a good fit for the data? c) Use the regression equation to estimate the percentage of deaths followed by cremation in 2016.

9 Section 1.5 Linear Equations, Functions, Zeros, and Applications 9 Solving Linear Equations The same number may be added or subtracted to both sides of an equation without changing the solution set. Both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution set. Example 1 Solve: 3(5 + 2x) = 4 2(x + 3) Motion Formula The distance d traveled by an object moving at rate r in time t is given by d = r t Simple-Interest Formula The simple interest I on a principal of P dollars at interest rate r for t years is given by I = Prt Five Steps for Problem Solving Familiarize yourself with the problem situation. Make a drawing Write a list Assign variables Organize into a chart or table Find further information Guess or estimate the answer Translate to mathematical language or symbolism. Carry out some type of mathematical manipulation. Check to see whether your possible solution actually fits the problem situation. State the answer clearly. Example 2 The average salary of a landscape architect for the federal government is $80,830 per year. This is about 38.5% higher than the yearly salary of a private-sector landscape architect. (Source: Bureau of Labor Statistics) Find the salary of a privatesector landscape architect. Example 3 Juliet has a choice between receiving an $1800 monthly salary from Pearson s Furniture or a base salary of $1600 and a 4% commission on the amount of furniture she sells during the month. For what amount of sales will the two choices be equal? Example 4 In triangle ABC, angle B is twice as large as angle A. Angle C measures 20 more than angle A. Find the measures of the angles.

10 10 Example 5 A truck enters a highway driving 60 mph. A car enters the highway at the same place 11 minutes later and drives 69 mph in the same direction. From the time the car enters the highway, how long will it take the car to pass the truck? Round to the nearest minute. Example 6 Dimitri s two student loans total $9000. One loan is at 5% simple interest and the other is at 6% simple interest. At the end of one year, Dimitri owes $492 in interest. What is the amount of each loan? Zeros of Functions A zero of a function is an x-value that makes the function, or y-value, equal. If x = a is a zero of a function, then (a, 0) is an x-intercept. Example 7 Use the given graph to find a) the x-intercept b) the zero of the function

11 Section 1.6 Solving Linear Inequalities 11 Solving Linear Inequalities When both sides of an inequality are multiplied or divided by a negative number, we must the direction of the inequality. Example 1 Solve and graph the solution set: 6(x 3) 7(x 2) Example 2 Solve and graph the solution set: Compound Inequalities (inequalities joined by the words and or or ) A conjunction contains the word and. -7 < 3x + 5 and 3x can be combined as -7 < 3x A disjunction contains the word or. 3x or 3x + 6 > 12 Example 3 Solve and graph the solution set: -3 < 1 2x 3 Example 4 Solve and graph the solution set: 4x 5-3 or 4x 5 > 3 Applications of Linear Inequalities Example 5 Johnson Catering charges $100 plus $30 per hour to cater an event. Catherine s Catering charges $50 per hour. For what length of time does it cost less to hire Catherine s Catering?