GRE 502 & GRE 503 LESSON/NOTES CRS SKILL GRE 502 GRE503 Period____________ LEVEL Level 1 – ALL students must attain mastery at this level Level 2 – MOST students will attain mastery of the focus skill in isolation. Level 3 – SOME students will attain mastery of focus skill with other skills Level 4 – SOME students will attain mastery of focus topics covered in a more abstract way Level 5 – FEW students will attain mastery of the extension skill. Name_________________________________________ DESCRIPTION GRE 401 Locate points on a coordinate plane GRE 403 Exhibit knowledge of slope GRE 502 Determine the slope of a line from points or equations GRE 503 Match linear graphs with their equations GRE 601 Interpret and use information from graphs in the coordinate plane GRE 604 Use properties of parallel and perpendicular lines to determine an equation of a line or coordinates of a point VOCABULARY Slope, X-­‐Intercept, Y-­‐Intercept, Slope Intercept Form, Point Slope Form, Standard Form, Horizontal Line, Vertical Line, Perpendicular Lines, Parallel Lines REQUIRED SKILL TO MASTER Identifying slope of a line*(algebra) Additional Practice Topics on KHAN Graphing Linear Equations
Solve for the x-intercept
Solve for the y-intercept
Slope intercept form
Graphing systems of equations
Level 1 1 Plotting points on a coordinate plane. ⎛ + right , + up ⎞
In mathematics, you use an ordered pair of number to describe where a point is ⎜
- down⎟⎠
⎝ - left
on a coordinate plane. 1. Plot the following: 6
A (-­‐6, 2) B (6, 1) 4
C (-­‐5, -­‐4) 2
D (3, -­‐1) E (0, 2) -5
5
F (-­‐3, 0) -2
-4
-6
2. If a house painter leans the top of his ladder against the wall of a house 10 feet above the ground and notices that the bottom of the ladder is approximately 4 feet from the base of the house, what is the slope of the painters ladder? 3. Would you feel comfortable climbing a ladder with the following slopes? Why or why not? 8
12
3
24
51
a) b)
c)0
d)
e)
f) 3
1
10
7
5
2 ACTIVITY
Suppose that you work part-­‐time in a music store. You earn $40 per week plus a 10% commission on all of the sales you make. 1. What does it mean to earn a commission? 2. You can represent this relationship between weekly sales and weekly wages by making a table. Complete the table below: Weekly sales, s(in dollars) Weekly wages, w (in dollars) 100 40+0.10(100)=50 200 300 400 500 3. What observations can you make about successive entries in the weekly sales column? In the weekly wages column? 4. You can represent each row in the table as an ordered pair (s,w). Plot each ordered pair, and connect the first and last points with a straightedge. Does each point that you plotted appear to be contained in this line segment? 5. Write an equation to represent the relationship between s and w. 3 4. A plumber charges a base fee of $55 for a service call plus $35 per hour for all hours worked during the call. a. Make a table of the total charge for 1, 2, 3, and 4 hours worked. b. Graph the points represented by your table. 18
Hours Worked 1 2 3 4 16
14
Total Charge 12
10
8
6
4
c. Write a linear equation to model this situation. d. Find the charge for 5 hours of work. 5. Louis started a savings account with $310. After 4 weeks, he had $350 dollars, and after 9 weeks, he had $400 a. Express the information given in this problem as a set of three-­‐ordered pairs. b. Use any two points above to find the rate of change of money in his savings account per week? 4 6. Make a drawing of the following slopes. 2
a)
3
3
b)
5
7
c)
2
11
d)
3
e)0
f)no slope
7. On the grid below plot each pair of points, draw the segment that joins them and then find the slope. 16
a) (8,6) (10,9) b) (6,0) (3,1) c) (3,5) (6,8) d) (2,3) (1,6) e) ( 8,9) (9,14) f) f) (1,14)(5,9) 14
12
10
8
6
4
2
5
10
-2
5 SLOPE INTERCEPT FORM The slope intercept form of a linear equation has the following form where the equation is solved for y in terms of x: y = a + bx b = the slope (rate of change) a = a constant term. It is the y intercept, the place where the line crosses the y-­‐axis. 8. Determine the slope of the following equations: a) y = −7x + 3
b) y = −5 − 9x
c) y =
−5 −6
+
x
4
7
d) y =
−5
−6
x+
4
7
Level 2 9. Find the slope and y-­‐intercept of each equation. If necessary convert the equations to slope-­‐
intercept form: y = a + bx. a) y = −5x +1 b) 4y = 3x + 8 −
c) 2y + 3x = 2 6 10. Write the equation of each line in the graphs below. 4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
11. Find the slope of the line that passes through the following points: (-­‐1, 6) and (3, 10) 12. Find the slope of the line that passes through the following points: (8, 3) and (-­‐7, -­‐2) 13. Graph each of the following: −2
y=
x−2
5
y = 5 + 3x 7 POINT SLOPE FORM The point slope form of a linear equation has the following form: y − y1 = b(x − x1 ) or y = y1 + b(x − x1 ) b = the slope (rate of change) ( x1, y1 ) = any point on the line 14. Find a Point-­‐Slope equation for a line containing the given point and having the given slope. a. (4, -­‐3), m = -­‐1 b. (-­‐7, 2), m = 3 c. (6, -­‐2), m = -­‐3 15. Write the equation of each line below, using the point-­‐slope form. a. b. 6
6
4
4
2
2
-5
-5
5
5
-2
-2
-4
-4
-6
-6
8 STANDARD FORM (GENERAL FORM) The standard form (general form) of a linear equation has the following form: Ax + By = C A,B, and C are integers 9 16. The lines below are given in standard form. Find the x-­‐ and y-­‐intercepts, a third point and then graph the line. a) 6x + 3y = 12
x-intercept=_______
y-intercept =_______
Point=____________
b) x – 2y = 6
c) 3x + y = 3
x-intercept=_______
x-intercept=______
y-intercept =______
y-intercept =______
Point=____________ Point=____________ 10 17. Write the equation 6x-­‐9y+45 =0 in slope-­‐intercept form. (Solve for y) Type of line Horizontal
Vertical
Slope Zero Equation Y = # Undefined or No
Slope X=# 18. Bear Mountain appears from a distance like the sketch below. a) What is the slope of the top of the mountain? b) What is the slope of the mountain to your right in the drawing? 19. Plot and label the following points: J (1,0) H (0,7) F (-­‐4, -­‐7) D ( -­‐6, 0) I (4, -­‐7) G (-­‐10, -­‐3) E (10, -­‐9) 20. Find the slope of each line: JD DI FG 3) Write the equation for each line: JD EG 10
8
6
4
2
-10
-5
5
10
-2
-4
-6
-8
-10
FH 11 Level 3 21. Graph and label the following vertical and horizontal lines. Then, determine the area of the rectangle enclosed by the lines. x = −2 x = 4 y = −3 y = 5 22. Graph and label the following lines. Then, determine the area of the triangle enclosed by the lines. Remember to solve for y in the equation of the slant line. x = 3 y = −2 3y − 2x = 6 12 23. For each situation find a graph and an equation that could represent the situation. Explain how you made each choice. A. B. Graph _____. Explanation: Equation______. Explanation: C. Graph _____. Explanation: Equation______. Explanation: D. Graph _____. Explanation: Equation______. Explanation: Graph _____. Explanation: Equation______. Explanation: 1. 2. 3. a. c. b. 4. d. 13 Level 4 24. Snow is falling at a rate of ½ inch per hour. Before the snowstorm began, there was already 6 inches on the ground. Write an equation in slope-­‐intercept form modeling the depth of the snow y(in inches) on the ground after x hours. 25. You have $50 in your savings account at the beginning of the year. Each month you save $30. a. Write an equation, in slope-­‐intercept form, which models the amount of money s (in dollars) in your savings account after m months. ________ = ________ m + ________
Total
Savings
Start
per Month
value
14
12
10
b. Graph the equation that models situation on the graph below. c. Use your equation to predict how much money you will have 18 months. D
o
l
l
a
r
s
this 8
6
in 4
2
2
Months
4
6
8
10
14 26. Amanda is trying to save $1,500 by the time she goes to college in two years. The table below shows the amount of money she has in her savings account each month. Months Amount Saved ($) a. Plot the points on the graph below. 0 100 5 350 10 600 b. Draw a straight line 15 850 through the points on the graph. Is the amount of money Amount
Amanda has Saved
increasing or decreasing compared to the money she had the month before? Months
c. Find the amount of money Amanda saved each month. d. Based on the table and graph, will Amada have saved her goal of $1500? 13. Burlington High School has made a donation pledge to a Love Inc. They have already donated $350 and will continue to donate $2 for each student who also signs a pledge. Write an equation in slope-­‐intercept form modeling the amount of the donation d(in dollars) for the number of student, s, who sign the pledge. 14. Pam and Kaitlin both wrote an equation for the following graph. Pam:
y = 3x +1
Kaitlin:
1
y = x +1
3
Who is correct? Explain the error made by the person with the incorrect answer. 15 Level 5 This exercise will investigate the slopes of parallel and perpendicular lines. The investigation will be done first on paper and then verified on Geometers Sketch Pad or Graphing Caculator as a whole group. 16 17 Whole Group 5. Using Geometr’s Sketchpad or graphing calculator to verify your conjecture. Create four different pairs of parallel lines. Then measure the slopes. Record the results below. 18 19 ______________________________________________________________________________________________________ 5. How do the slopes of the lines in the third pair of lines fit our conjecture?__________________ ______________________________________________________________________________________________________ 6. How do the slopes of the lines in the fifth pair of lines fit your conjecture?__________________ ______________________________________________________________________________________________________ ______________________________________________________________________________________________________ 20 7. As a whole group use Geometr’s Sketchpad or graphing calculator to verify your conjecture. Create four different pairs of perpendicular lines. Use the slope tool to find the slope of each line. 27.
Find the slope of a line parallel to and the slope of a line that is perpendicular to the graph of each of the following equations. Equation y=
−3
x − 7 4
y = 12 + 5x 16x-­‐32y=160 3
y − 8 = (x − 2) 5
Slope of a line parallel Slope of a line perpendicular 21 22 Find the slope of a line parallel to the following lines: (show work if possible) 28.
y = x – 1 29.
7x – y = 4 a. -­‐2/3 a. -­‐1/7 b. -­‐2 b. -­‐7 c. 2 c. 1/7 d. -­‐1 d. 7 e. 1 e. 1 Find the slope of a line perpendicular to the following lines: (show your work) !
30.
8x – 5y = 15 31.
y = − !x + 3 a. 5/8 a. -­‐5/3 b. 8/5 b. 5/3 c. -­‐8/5 c. -­‐3/5 d. -­‐5 d. -­‐1/3 e. -­‐5/8 e. 3/5 32.
Write an equation for the line that passes through the point (3, -­‐3) and is perpendicular to the line whose equation is y = -­‐3x. 33. Write an equation in slope-­‐intercept form for the line that contains the point (-­‐2,7) and is parallel to the graph of y= 3x − 4. 23