Algebra II Note workbook Chapter 2 Name
Algebra II: 2-1 Relations and Functions The table shows the average lifetime and maximum lifetime for some animals. This data can be written as. The ordered pairs of the data are. The first number in each ordered pair is the and the second number is the. We represent ordered pairs by. When we graph, we graph on a coordinate system with. The two axis coordinate system is called the. The horizontal axis is labeled the and the vertical axis is labeled the. The point at Which the two axes meet is called the and its coordinates are. The four regions divided by the coordinate axes are called. We ALWAYS write ordered pairs in the form. Ex. 1: Label all of the important parts of the coordinate axis below.
A is a set of ordered pairs. The is the set of all first coordinates ( ) and the is the set of all second coordinates ( ). Ex. 2: Consider the set: {(-1, 3), (2,4), (3, 1), (8, 6)}. a) Find the domain of the set. b) Find the range of the set. A is a special type of relation in which each element on the domain is paired with one element of the range. We often draw as a way to represent functions. A special function is a function. In this kind of function each element of the range is paired with exactly one element of the domain. Ex. 3: State the domain and range of the relation shown in the graph. Is the graph a relation?
Ex. 4: Is the set {(9, 3), (9, -3), (4, 2), (4, -2)} a function? Construct a mapping of this relation. If a relation is graphed, it is very easy to tell if a graph is a function. We use the. Ex. 5: Are the following relations functions? a) b) c) d)
Ex. 6: The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Graph this information and determine whether it represents a function. So far, we have learned three ways that relations and functions can be represented: 1) 2) 3) The fourth way to represent relations and functions is by writing an. YOU NEED TO BE ABLE TO MOVE BACK AND FORTH BETWEEN ALL FOUR REPRESENTATIONS OF FUNCTIONS!!!!!!!!!!!!!!!!!! The of an equation in x and y are the set of ordered pairs that make the equation. Consider the equation 2 6. What can x be?. So, we say that the domain is. Ex. 7: Graph the relation by represented 2 1.
Ex. 7 con t.: b) Find the domain and range of 2 1. b) Is this relation a function? When an equation represents a function, the variable, usually x, whose values make up the domain is called the. The other variable, usually y, is called the because its values depend on x. When equations are represented by functions, we usually write the equation in. For example, we can write the equation y = 2x +1 as f (x) = 2x +1. The symbol f (x) replaces the y. The f is just the of the function. It is NOT a variable that is multiplied by x. Suppose you want to find the value in the range that corresponds to the element 4 in the domain of the function. This is written as f (4). The value f (4) is found by substituting 4 for each x in the equation. So, f (4) =. Ex. 8: Given 2 and 0.5 5 3.5, find each value. a) 3 b) 2.8 c) 3
HW: Day One 1) a) b) c) d) e) f) 2) Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. a. 2,1, 3,0, 1,5 b. 4,5, 6,5, 3,5 c. 5 d. 3 3) For question 3, use the table that shows a company s stock price in recent years. a. Write a relation to represent the data. b. Graph the relation and tell if it is a function. c. Identify the domain and range.
HW: Day Two: 1) Find each value if 3 5 and. a. 3 b. 3 c. d. e. f. 5 2) Find the value of 3 2 when 2. 3) What is 4 if 5? 4) If 2 5, then 0 a. 0 b. 5 c. 3 d. 5) If, then 1 a. 1 b. 1 c. 2 1 d.
Algebra II: 2-2 Linear Equations Definition: A is an equation that has no operations other than,, and multiplication of a variable by a constant. The variables MAY NOT be together or appear in a. All variables have exponents of. *******KEY IDEA: The graph of a linear equations is ALWAYS a. Linear Equations Not Linear Equations 5 3 7 9 6 3 15 7 4 8 5 1 1 2 1 Def: A is a function whose ordered pairs satisfy a linear equation. Any linear function can be written in the form, where m and b are real numbers. Ex. 1: State whether the function is a linear function. a) 10 5 b) 5 c), 2
Ex. 2: The linear function 1.8 32 can be used to find the number of degrees Fahrenheit, f, that are equivalent to a given number of degrees Celsius, C. a) On the Celsius scale, normal body temperature is 37 o C. What is the normal body of temperature in degrees Fahrenheit? b) There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree? Any linear equation can be written in. The standard form of a linear equation is, where A, B, and C are integers whose greatest common factor is 1. Ex. 3: Write each equation in standard form. Identify A, B, and C. a. 3 9 b. 2 1 c. 8 6 4 0
Special points on a line. One of the quickest ways to graph a line is to identify where the line crosses the and the. The point where the line crosses the x-axis is called the and the point where the line crosses the y-axis is called the. To find the x- and y- intercepts: To find the x-intercept, let y = 0. To find the y-intercept, let x = 0. Ex. 4: Find the x-intercept and the y-intercept of the graph of 3 4 12 0. Then graph the equation. HW: 1) State whether the function is linear. a. 4 b. 1.1 2 c. 5 d. 3 5 e. 4 f. 2 4 5 2) Which of the following equations is NOT linear? a. 9 7 b. 5 0 c. 3 1 d. 2 3) Which of the following equations is a linear function? a. 3 b. 1 c. 3 d. 9
4) When a sound travels through water, the distance in meters that the sound travels in seconds is given by the equation 1440. a. How far does a sound travel underwater in 5 seconds? b. In air, the equation is 343. Does sound travel faster in air or water? Explain. 5) Write each equation in standard form. a. 3 4 b. 12 c. 4 5 d. 5 10 25 e. 6 f. 0.5 3 6) Find the and intercepts of the graph of each equation and then graph the equation. a. 5 3 15
b. 2 6 12 c. 2 5 10 0 d. 4 2 e. 2 f. 8
7. Suppose the temperature T ( o C) below the Earth s surface is given by 35 20, where d is the depth (km). a. Find the temperature at a depth of 2 kilometers. b. Find the depth if the temperature is 160 o C. c. Graph the linear function. 8. The Jackson Band Boosters sell beverages for $1.75 and candy for $1.50 at home games. Their goal is to have total sales of $525 for each game. a. Write an equation that is a model for the different numbers of beverages and candy that can be sold to meet the goal. b. Graph your equation. c. Does this equation represent a function? d. If they sell 100 beverages and 200 pieces of candy, will the Band Boosters meet their goal?
Algebra II: 2-3 Slope Definition: The of a line is the ratio of the change in y-coordinates to the corresponding change in x-coordinates. Suppose a line passes through the points, and,. Then, slope =. Ex. 1: Find the slope of the line that passes through (1, 3) and (-2, -3). Then graph the line. Ex. 2: Graph the line passing through (1, -3) with a slope of.
The slope of a line tells the direction in which the line or. In real life and in science classes, slope is often referred to as the. It measures how much one quantity changes on average relative to the change in another quantity, often time. Examples of common rates of change include:. Ex. 3:
Parallel and Perpendicular Lines: Using a graphing calculator, draw the graphs of 3 2, 3 2, and 3 5 on the same coordinate plane. What do you notice about all 3 of these graphs? Can you make a conjecture based on what you see?
Ex. 4: If line a is parallel to the line 3 4 8, what is the slope of line a? Ex. 5: Graph the line through 1, 2 that is parallel to the line with equation 2. We say that the slopes are of each other. When you multiply the slopes of two perpendicular lines, the product is always.
Ex. 6: Suppose line l is perpendicular to the line given by 2 4. What is the slope of line l? Ex. 7: Graph the line through (2, 1) that is perpendicular to the line with equation 2 3 3.
HW: Day One 1) Find the slope of the line that passes through each pair of points. a. 6,1, 8, 4 b. 6, 5, 4,1 c. 7,8, 1,8 d. 4, 1.5, 4,4.5 2) Determine the value of so that the line through 6, and 9,2 has slope. 3) Graph the line passing through the given point with the given slope. a. 2,6, b. 3, 1, c. 3, 4, 2 d. 1,2 3
4) Refer to the graph that shows the number of CD s and cassette tapes shipped by manufacturers to retailers in recent years. a. Find the average rate of change of the number of CD s shipped from 1991 to 2000. b. Find the average rate of change of the number of cassette tapes shipped from 1991 to 2000. c. Interpret the sign of your answer to part b). 5) Mr. And Mrs. Wellman are taking their daughter to college. The table shows their distance from home after various amounts of time. a. Find the average rate of change of their distance from home between 1 and 3 hours after leaving home. b. Find the average rate of change of their distance from home between 0 and 5 hours after leaving home. c. C. What is another word for rate of change in this situation?
HW: Day Two: 1) Graph the line that satisfies each set of conditions. a. Passes through 2,2, parallel to a line whose slope is 1. b. Passes through 4,1, perpendicular to a line whose slope is. c. Passes through 3,3, perpendicular to the graph of 3. d. Passes through 2, 1, parallel to graph of 2 3 6.
Section 2 4: Writing Linear Equations Date: Forms of Equations Consider the following graph. The line passes through and. Notice that is the y-intercept of. You can use these two points to find the slope of. Find the slope: Now solve your new equation for y. This form is called the. Example #1: Write an Equation Given Slope and a Point Write an equation in slope-intercept form for the line that has a slope of through (5, - 2). 3 5 and passes 1
Point-slope form: use this form to find an equation of a line when you are given the of two on a line Example #2: Write an Equation Given Two Points Write an equation of the line through (2, - 3) and ( - 3, 7). When changes in real-world situations occur at a equation can be used as a for describing the situation., a linear Example #3: Write an Equation for a Real-World Situation Sales As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. a) Write a linear equation to model this situation. b) What are Ms. Stock s daily salary and commission rate? c) How much would Jean make in a day if her sales were $500? 2
Parallel and Perpendicular Lines The and forms can be used to find equations of lines that are or to given lines. Example #4: Write an Equation of a Perpendicular Line Write an equation for the line that passes through (3, - 2) and is perpendicular to the lien whose equation is y = 5 x + 1. HW: 1) State the slope and y-intercept of the graph of each equation. a. 4 b. c.2 4 10 d. 3 5 30 0 e. 7 f. 2) Write an equation in slope-intercept form for each graph. a. b. 3
c. d. 3) Write an equation in slope-intercept form for the line that satisfies each set of conditions. a. slope 3, passes through 0, 6 b. slope 0.25, passes through 0,4 c. slope, passes through 1,3 d. slope, passes through 5,1 e. x-intercept 4, y-intercept 4 f. passes through 2, 5, perpendicular to 7 4) 4
Date: Section 2 5: Modeling Real-World Data: Using Scatter Plots Real Numbers Data with two variables, is called. A set of bivariate data graphed as ordered pairs in a coordinate plane is called a. A scatter plot can show whether there is a between the data. Example #1: Draw a Scatter Plot Education The table shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data. Prediction Equations When you find a line that closely approximates a set of, you are finding a for the data. An equation of such a line is often called a because it can be used to predict one of the variables given the other variable. To find a line of fit and a prediction equation for a set of data, select that appear to represent the data well. This is a matter of personal judgment, so your line and prediction equation may be different from someone else s. 5
Example #2: Find and Use a Prediction Equation Education Refer to the data in Example #1. a) Draw a line of fit for the data. How well does the line fit the data? b) Find a prediction equation. What do the slope and y-intercept indicate? c) Predict the percent in 2010. d) How accurate is the prediction? 6
HW: 1) 2) Whether you are climbing a mountain or flying in an airplane, the higher you go, the colder the air gets. The table shows the temperature in the atmosphere at various altitudes. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use your prediction equation to predict the missing value. 3) As more channels have been added, cable television has become attractive to more viewers. The table shows the number of U.S. households with cable service in recent years. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use your prediction equation to predict the missing value. 7
Date: Section 2 6: Special Functions Step Functions, Constant Functions, and the Identity Function The cost of postage to mail a letter is a of the weight of the letter. But the function is not. It is a special function called a. The graph of a step function is not. It consists of line segments or rays. The, written, is an example of a step function. The symbol means the greatest integer less than or equal to x. For example, and because. Study the table and graph below. Example #1: Step Function Psychology One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation. 8
You ve learned that the slope-intercept form of a linear function is, or in functional notation,. When m=0, the value of the function is for every x value. So, f(x) = b is called a. The function f(x) = 0 is called the. Example #2: Constant Function Graph g ( x) = 3. First make a table of values. Another special case of slope-intercept form is,. This is the function. The graph is the line through the with slope 1. Since the function does not change the input value, is called the. Absolute Value and Piecewise Functions Another special function is the,. 9
The absolute value function can be written as. A function that is written using two or more is called a. Recall that a family of graphs is a group of graphs that displays one or more similar. The parent graph of most absolute value functions is. Example #3: Absolute Value Functions Graph f ( x) = x 3 and g ( x) = x + 2 on the same coordinate plane. Determine the similarities and differences in the two graphs. To graph other piecewise functions, examine the in the definition of the function to determine how much of each piece to include. Example #4: Piecewise Function x 1if x 3 Graph f ( x) =. 1if x > 3 Identify the domain and range. 10
Example #5: Identify Functions Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a) b) HW: 1) a. b. c. d. e. f. 11
2) Bluffton High School chartered buses so the student body could attend the girls basketball state tournament games. Each bus held a maximum of 60 students. Draw a graph of a step function that shows the relationship between the number of students x who went to the game and the number of buses y that were needed. 3) Graph each function. Identify the domain and range. a. 3 b. 2 c. 3 d. 4, 3 e. 2, 3 12
Section 2 7: Graphing Inequalities Date: Graph Linear Inequalities A linear inequality resembles a linear equation, but with an inequality symbol instead of an. For example, is a linear inequality and is the related linear equation. The graph of the inequality is the region. Every point in the shaded region satisfies the inequality. The graph of is the of the region. It is drawn as to show that points on the line satisfy the inequality. If the inequality symbol were < or >, then points on the boundary would not satisfy the inequality, so the boundary would be drawn as a You can graph an inequality by following these steps. Step 1 Determine whether the boundary should be or. Graph the boundary Step 2 Choose a not on the boundary and test the inequality. Step 3 If a inequality results, shade the region containing your test point. If a inequality results, shade the other region. Example #1: Dashed Boundary Graph x 2 y < 4. 13
Example #2: Solid Boundary Business A mail-order company is hiring temporary employees to help in their packing and shipping departments during their peak season. a) Write an inequality to describe the number of employees that can be assigned to each department if the company has 20 temporary employees available. b) Graph the inequality. c) Can the company assign 8 employees to packing and 10 employees to shipping? 14
Graph Absolute Value Inequalities Graphing absolute value inequalities is similar to graphing inequalities. The inequality symbol determines whether the boundary is or, and you can test a point to determine which region to. Example #3: Absolute Value Inequality Graph y x 2. HW: Graph each inequality 1) 5 2) 3 3 3) 6 2 4) 5 4 6 3 15