Lesson 4 Contents Example 1 Write an Equation Given Slope and a Point Example 2 Write an Equation Given Two Points Example 3 Write an Equation for a Real-World Situation Example 4 Write an Equation of a Perpendicular Line Lesson 4 Contents

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Slope-Intercept Form of an Equation The slope-intercept form of the equation of a line is written y = mx + b, where m is the slope of the line and b is the y-intercept of the line. Examples of equations written in slope-intercept form are: y = 3x + 2 y = -2x + 5 y = ½ x - 4

Write an equation in slope-intercept form for the line that has a slope of and passes through (5, –2). Slope-intercept form Simplify. Example 4-1a

Example 4-1b Add 3 to each side. Answer: The y-intercept is 1. So, the equation in slope-intercept form is Example 4-1b

Write an equation in slope-intercept form for the line that has a slope of and passes through (–3, –1). Answer: Example 4-1c

m = 4, passes through the origin Try These Write an equation in slope-intercept form for the line with the given slope passing through the given point. m = 3, P(0, -6) m = -1/2, P(1, 3) m = -0.5, P(2, -3) m = 4, passes through the origin

m = 4, passes through the origin Try These Write an equation in slope-intercept form for the line with the given slope passing through the given point. m = 3, P(0, -6) m = - ½, P(1, 3) m = -0.5, P(2, -3) m = 4, passes through the origin y = 3x – 6 y = - ½ x + 3 ½ y = -0.5x – 2 y = 4x

Example 4-2a Multiple-Choice Test Item What is an equation of the line through (2, –3) and (–3, 7)? A B C D Read the Test Item You are given the coordinates of two points on the line. Notice that the answer choices are in slope-intercept form. Example 4-2a

Example 4-2b Solve the Test Item First, find the slope of the line. Slope formula Simplify. The slope is –2. That eliminates choices B and C. Example 4-2b

Then use the slope-intercept form of an equation to find b. Y = mx + b Slope-intercept form -3 = -2 (2) + b Replace x and y with values from point given and use slope -3 = -4 + b Simplify. 1 = b Simplify. Answer: D Example 4-2c

Example 4-2d Multiple-Choice Test Item What is an equation of the line through (2, 5) and (–1, 3)? A C B D Answer: C Example 4-2d

Try These Write an equation in slope-intercept form for the line passing through the given points. A(-2, 5), B(2, 1) A(-4, 0), B(3, 0) A(-2, -3), B(0, 0) A(8, 2), B(5, 2)

Try These Write an equation in slope-intercept form for the line passing through the given points. A(-2, 5), B(2, 1) A(-4, 0), B(3, 0) A(-2, -3), B(0, 0) A(8, 8), B(5, 2) y = 0 y = -x + 3 y = 3/2x y = 2x – 8

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. Write a linear equation to model this situation. Let x be her sales and let y be the amount of money she makes. Use the points (100, 58) and (300, 78) to make a graph to represent the situation. Example 4-3a

Slope formula Simplify. Example 4-3b

Now use the slope and either of the given points with the point-slope form to write the equation. Distributive Property Add 58 to each side. Answer: The slope-intercept form of the equation is Example 4-3c

Example 4-3d What are Ms. Stock’s daily salary and commission rate? The y-intercept of the line is 48. The y-intercept represents the money Jean would make if she had no sales. Thus, $48 is her daily salary. The slope of the line is 0.1. Since the slope is the coefficient of x, which is her sales, she makes 10% commission. Answer: Ms. Stock’s daily salary is $48, and she makes a 10% commission. Example 4-3d

Example 4-3f How much would Jean make in a day if her sales were $500? Find the value of y when Use the equation you found in Example 3a. Replace x with 500. Simplify. Answer: She would make $98 if her sales were $500. Example 4-3f

Sales The student council is selling coupon books to raise money for the Humane Society. If the group sells 200 books, they will receive $150 dollars. If they sell 500 books, they will make $375. a. Write a linear equation to model this situation. b. Find the percentage of the proceeds that the student council receives. c. If they sold 1000 books, how much money would they receive to donate to the Humane Society? Answer: Answer: 75% Answer: $750 Example 4-3h

Parallel & Perpendicular Lines Parallel lines have the same slope Ex) Slope of line A is 3 Slope of line B is 3 Perpendicular lines have opposite reciprocal slopes Ex) Slope of line A is 2/3 Slope of line B is –3/2

Write an equation for the line that passes through (3, –2) and is perpendicular to the line whose equation is The slope of the given line is –5. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular line is Use the point-slope form and the ordered pair (3, –2) to write the equation. Example 4-4a

Example 4-4b Point-slope form Distributive Property Subtract 2 from each side. Answer: An equation of the line is Example 4-4b

Write an equation for the line that passes through (3, 5) and is perpendicular to the line whose equation is Answer: Example 4-4c

Assignment: Page 78-79 #20, 26, 32, 36

End of Lesson 4

Lesson 5 Contents Example 1 Draw a Scatter Plot Example 2 Find and Use a Prediction Equation Lesson 5 Contents

Education The table below shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data. 13 14 15 18 20 Percent 12 9 6 3 Years Since 1985 Source: U.S. News & World Report Graph the data as ordered pairs, with the number of years since 1985 on the horizontal axis and the percentage on the vertical axis. Example 5-1a

Safety The table below shows the approximate percent of drivers who wear seat belts in various years since 1994. Make a scatter plot of the data. 73 71 68 69 64 61 58 57 Percent 7 6 5 4 3 2 1 Years Since 1994 Source: National Highway Traffic Safety Administration Example 5-1b

Education The table and scatter plot below show the approximate percent of students who sent applications to two colleges in various years since 1985. Draw a line of fit for the data. How well does the line fit the data? 13 14 15 18 20 Percent 12 9 6 3 Years Since 1985 Source: U.S. News & World Report The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points. Example 5-2a

Education The table and scatter plot below show the approximate percent of students who sent applications to two colleges in various years since 1985. Draw a line of fit for the data. How well does the line fit the data? 13 14 15 18 20 Percent 12 9 6 3 Years Since 1985 Source: U.S. News & World Report Answer: Except for (6, 15), this line fits the data fairly well. Example 5-2b

Find a prediction equation. What do the slope and y-intercept indicate? Find an equation of the line through (3, 18) and (15, 13). Begin by finding the slope. Slope formula Substitute. Simplify. Example 5-2c

Example 5-2d Point-slope form Distributive Property Add 18 to each side. Answer: One prediction equation is The slope indicates that the percent of students sending applications to two colleges is falling at about 0.4% each year. The y-intercept indicates that the percent in 1985 should have been about 19%. Example 5-2d

Example 5-2e Predict the percent in 2010. The year 2010 is 25 years after 1985, so use the prediction equation to find the value of y when Prediction equation Simplify. Answer: The model predicts that the percent in 2010 should be about 9%. Example 5-2e

Example 5-2f How accurate is this prediction? Answer: The fit is only approximate, so the prediction may not be very accurate. Example 5-2f

Safety The table and scatter plot show the approximate percent of drivers who wear seat belts in various years since 1994. a. Draw a line of fit for the data. How well does the line fit the data? 73 71 68 69 64 61 58 57 Percent 7 6 5 4 3 2 1 Years Since 1994 Source: National Highway Traffic Safety Administration Answer: Except for (4, 69), this line fits the data very well. Example 5-2g

b. Find a prediction equation. What do the slope and b. Find a prediction equation. What do the slope and y-intercept indicate? Answer: Using (1, 58) and (7, 73), an equation is y = 2.5x + 55.5. The slope indicates that the percent of drivers wearing seatbelts is increasing at a rate of 2.5% each year. The y-intercept indicates that, according to the trend of the rest of the data, the percent of drivers who wore seatbelts in 1994 was about 56%. Example 5-2h

c. Predict the percent of drivers who will be wearing c. Predict the percent of drivers who will be wearing seat belts in 2005. d. How accurate is the prediction? Answer: 83% Answer: Except for the outlier, the line fits the data very well, so the predicted value should be fairly accurate. Example 5-2i

Assignment: Page 84 #6-8

End of Lesson 5

Lesson 6 Contents Example 1 Step Function Example 2 Constant Function Example 3 Absolute Value Functions Example 4 Piecewise Function Example 5 Identify Functions Lesson 6 Contents

Step Functions A step function consists of lines or rays. It is not linear. Example: FITNESS A fitness center charges customers $7.50 per hour or any fraction thereof. Draw a graph that represents this situation.

Step Functions Explore: The total charge must be a multiple of 7.5, so the graph will be the graph of a step function. Plan: If the time spent at the center is greater than 0 hours, but less than or equal to 1 hour, then the charge will be $7.50. If the time is greater than 1 hour but less than or equal to 2 hours, then the labor cost is $15.00, and so on.

Step Functions Solve: Use the pattern of times and charges to make a table, where x is the number of hours spent at the center and C(x) is the total charge. Then draw the graph.

Psychology One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this solution. Explore The total charge must be a multiple of $85, so the graph will be the graph of a step function. Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on. Example 6-1a

Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph. 425 340 255 170 85 C(x) x Example 6-1b

Answer: Examine Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint. Example 6-1c

Sales The Daily Grind charges $1 Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation. Answer: Example 6-1d

Constant Functions A constant function is a function where the value of f(x) remains the same. The graph of a constant function is a horizontal line. Example: f(x) = –5 is a constant function since the value of f(x) is –5 regardless of what x is.

Constant Function By graphing the points in the chart we come up with the horizontal line below.

Example 6-2a Graph For every value of The graph is a horizontal line. Answer: g(x) = –3 x g(x) –2 –3 1 0.5 Example 6-2a

Graph Answer: Example 6-2b

Absolute Value Functions When graphing absolute value functions you can make a solution table. Be sure to choose positive and negative values in order to see both sides of the graph.

Absolute Value Functions The graph of the absolute value function f (x) = | x| is similar to the graph of f (x) = x except that the "negative" half of the graph is reflected over the x-axis. Here is the graph of f (x) = | x|:

Graph. and. on the same coordinate plane Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs. Find several ordered pairs for each function. x | x – 3 | 3 1 2 4 5 x | x + 2 | –4 2 –3 1 –2 –1 3 Example 6-3a

Example 6-3b Graph the points and connect them. Answer: The domain of both graphs is all real numbers. The range of both graphs is The graphs have the same shape, but different x-intercepts. The graph of g (x) is the graph of f (x) translated left 5 units. Example 6-3b

Graph. and. on the same coordinate plane Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs. The domain of both graphs is all real numbers. The graphs have the same shape, but different y-intercepts. The graph of g (x) is the graph of f (x) translated up 5 units. Answer: The range of is The range of is Example 6-3c

Piecewise Functions Example of a piecewise function is f(x) = A function that is written using two or more expressions. A piecewise defined graph is a graph whose behavior is defined in pieces. Along one portion of the domain, the graph may appear linear. On another portion, the graph may be parabolic. Many real-life situations are modeled with piecewise defined functions: Example of a piecewise function is f(x) =

Example 6-4a Graph Identify the domain and range. Step 1 Graph the linear function for Since 3 satisfies this inequality, begin with a closed circle at (3, 2). Example 6-4a

Example 6-4b Graph Identify the domain and range. Step 2 Graph the constant function Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right. Example 6-4b

Example 6-4c Graph Identify the domain and range. Answer: The function is defined for all values of x, so the domain is all real numbers. The values that are y-coordinates of points on the graph are all real numbers less than or equal to –2, so the range is Example 6-4c

Example 6-4d Graph Identify the domain and range. Answer: The domain is all real numbers. The range is Example 6-4d

Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer: Since this graph consists of different rays and segments, it is a piecewise function. Example 6-5a

Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer: Since this graph is V-shaped, it is an absolute value function. Example 6-5b

Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a. b. Answer: constant function Answer: absolute value function Example 6-5c

Refer to Page 93 in your textbooks. Complete #15-20. Try These Refer to Page 93 in your textbooks. Complete #15-20.

End of Lesson 6

Assignment: Page 94 #30, 32, 34, 36

Lesson 7 Contents Example 1 Dashed Boundary Example 2 Solid Boundary Example 3 Absolute Value Inequality Lesson 7 Contents

Graphing Inequalities If you can graph a straight line, you can graph an inequality!   After graphing the line, there are only two additional steps to remember.  (1) Choose a point not on the line and see if it makes the inequality true.  If the inequality is true, you will shade THAT side of the line -- thus shading OVER the point.  If it is false, you will shade the OTHER side of the line -- not shading OVER the point. (2)   If the inequality is LESS THAN OR EQUAL TO or GREATER THAN OR EQUAL TO, the line is drawn as a solid line.  If the inequality is simply LESS THAN or GREATER THAN, the line is drawn as a dashed line.

Example of Graphing an Inequality Example: Graph the following  inequality  y 3x - 1   1. Solve the equation for y (if necessary). 2.  Graph the equation as if it contained an = sign. 3.  Draw the line solid if the inequality is  or 4.  Draw the line dashed if the inequality is < or >  5.  Pick a point not on the line to use as a test point. The point (0,0) is a good test point if it is not on      the line. 6.  If the point makes the inequality true, shade that      side of the line.  If the point does not make the      inequality true, shade the opposite side of the line.

Example of Graphing an Inequality The point (-2,1) was chosen as the test point, because it can be clearly seen in the diagram. The easiest test point is usually (0,0] 1 3(-2) - 1 1 -6 -1 1 -7 false (shade the opposite side of the line)

Graph The boundary is the graph of Since the inequality symbol is <, the boundary will be dashed. Use the slope-intercept form, Example 7-1a

Example 7-1b Graph Test (0, 0). Original inequality true Shade the region that contains (0, 0). Example 7-1b

Graph Answer: Example 7-1c

Try These Graph each inequality. 1. x – 6y + 3 > 0 y > |4x|

Education The SAT has two parts Education The SAT has two parts. One tutoring company advertises that it specializes in helping students who have a combined score on the SAT that is 900 or less. Write an inequality to describe the combined scores of students who are prospective tutoring clients. Let x be the first part of the SAT and let y be the second part. Since the scores must be 900 or less, use the  symbol. Example 7-2a

The 2nd part are less than 1st part and together or equal to 900. x y  900 Answer: Example 7-2b

Example 7-2c Graph the inequality. Since the inequality symbol is , the graph of the related linear equation is solid. This is the boundary of the inequality. Example 7-2c

Example 7-2d Graph the inequality. Test (0, 0). Original inequality true Example 7-2d

Example 7-2eb Graph the inequality. Shade the region that contains (0, 0). Since the variables cannot be negative, shade only the part in the first quadrant. Example 7-2eb

Does a student with a verbal score of 480 and a math score of 410 fit the tutoring company’s guidelines? The point (480, 410) is in the shaded region, so it satisfies the inequality. Answer: Yes, this student fits the tutoring company’s guidelines. Example 7-2f

Class Trip Two social studies classes are going on a field trip Class Trip Two social studies classes are going on a field trip. The teachers have asked for parent volunteers to also go on the trip as chaperones. However, there is only enough seating for 60 people on the bus. a. Write an inequality to describe the number of students and chaperones that can ride on the bus. Answer: Example 7-2g

Example 7-2h b. Graph the inequality. Answer: c. Can 45 students and 10 chaperones go on the trip? Answer: Answer: yes Example 7-2h

Graph Since the inequality symbol is , the graph of the related equation is solid. Graph the equation. Test (0, 0). Original inequality Shade the region that contains (0, 0). true Example 7-3a

Graph Answer: Example 7-3b

End of Lesson 7

Assignment: Page 98 #18, 35, 36, 37

Assignment P 98 #14-26 every other even, 32, 35-37