Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Warm Up Solve each equation for y. 1. 6y – 12x = 24 2. –2y – 4x = 20 3. 2y – 5x = 16 4. 3y + 6x = 18 y = 2x + 4 y = –2x – 10 y = x + 8 5 2 y = –2x + 6

Review Skills For Chapter 11 Operations with Integers (to include fractions) Solving Multistep Equations and Inequalities i.e. solve 4m-5(m+2)=1 Solve Equations for One Variable (i.e. solve 5y-x = 4 for y) Solve Inequalities for One Variable

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Essential Question: Explain whether an equation is linear if three ordered pair solutions lie on a straight line but a fourth does not. Learn to identify and graph linear equations. Objective: 5.01a, 5.01b, 5.03 Develop an understanding of function with algebraic representations and identify relations as linear or nonlinear.

Vocabulary (Add to vocabulary folder) Course 3 11-1 Graphing Linear Equations Vocabulary (Add to vocabulary folder) A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. To find a solution that lies between two points (x1, y1) and (x2, y2), choose an x-value between x1 and x2 and find the corresponding y-value.

Graphing Linear Equations Insert Lesson Title Here Course 3 11-1 Graphing Linear Equations Insert Lesson Title Here Reading Math Read x1 as “x sub one” or “x one.” So, x2- x1 would be read, X sub 2 minus x sub 1 You will see this again very soon! (Hint: 11-2) We will be referencing ordered pairs to determine the value of the x’s and the y’s.

Common Error Alert! Students may incorrectly graph a point so that it is not collinear with the others. Encourage students to double check any points that are not collinear. (Collinear – points lying on the same line) In short, if you graph a point and it is NOT on the line, double check your work to ensure that you have not made a mistake!

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations If an equation is linear, a constant change in the x-value corresponds to a constant change in the y-value. The graph shows an example where each time the x-value increases by 3, the y-value increases by 2. Hey! This is Important! you will see this again very soon! 2 3 2 3 2 3

Additional Example 1A: Graphing Equations Course 3 11-1 Graphing Linear Equations Additional Example 1A: Graphing Equations Graph the equation and tell whether it is linear. A. y = 3x – 1 x 3x – 1 y (x, y) –2 –1 1 2 3(–2) – 1 –7 (–2, –7) 3(–1) – 1 –4 (–1, –4) 3(0) – 1 –1 (0, –1) 3(1) – 1 2 (1, 2) 3(2) – 1 5 (2, 5)

Additional Example 1A Continued Course 3 11-1 Graphing Linear Equations Additional Example 1A Continued The equation y = 3x – 1 is a linear equation because it is the graph of a straight line and each time x increases by 1 unit, y increases by 3 units.

Additional Example 1B: Graphing Equations Course 3 11-1 Graphing Linear Equations Additional Example 1B: Graphing Equations Graph the equation and tell whether it is linear. B. y = x3 x x3 y (x, y) –2 –1 1 2 (–2)3 –8 (–2, –8) (–1)3 –1 (–1, –1) (0)3 (0, 0) (1)3 1 (1, 1) (2)3 8 (2, 8)

Additional Example 1B Continued Course 3 11-1 Graphing Linear Equations Additional Example 1B Continued The equation y = x3 is not a linear equation because its graph is not a straight line. Also notice that as x increases by a constant of 1 unit, the change in y is not constant. x –2 –1 1 2 y –8 8 +7 +1 +1 +7

Additional Example 1C: Graphing Equations Course 3 11-1 Graphing Linear Equations Additional Example 1C: Graphing Equations Graph the equation and tell whether it is linear. C. y = – 3x 4

Additional Example 1 Continued Course 3 11-1 Graphing Linear Equations Additional Example 1 Continued The equation y = – is a linear equation because the points form a straight line. Each time the value of x increases by 1, the value of y decreases by or y decreases by 3 each time x increases by 4. 3x 4 3

Additional Example 1D: Graphing Equations Course 3 11-1 Graphing Linear Equations Additional Example 1D: Graphing Equations Graph the equation and tell whether it is linear. D. y = 2 x 2 y (x, y) –2 –1 1 2 2 (–2, 2) 2 2 (–1, 2) 2 2 (0, 2) 2 2 (1, 2) 2 2 (2, 2) For any value of x, y = 2.

Additional Example 1D Continued Course 3 11-1 Graphing Linear Equations Additional Example 1D Continued The equation y = 2 is a linear equation because the points form a straight line. As the value of x increases, the value of y has a constant change of 0.

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Try This: Example 1A Graph the equation and tell whether it is linear. A. y = 2x + 1 x 2x + 1 y (x, y) –2 –1 1 2 2(–2) + 1 –3 (–3, –3) 2(–1) + 1 –1 (–2, –1) 2(0) + 1 1 (–1, 1) 2(1) + 1 3 (0, 3) 2(2) + 1 5 (2, 5)

Try This: Example 1A Continued Course 3 11-1 Graphing Linear Equations Try This: Example 1A Continued The equation y = 2x + 1 is linear equation because it is the graph of a straight line and each time x increase by 1 unit, y increases by 2 units.

Group Work The next 5 problems (3 equations & 2 word problems) will be done in groups of 2. Each group will write and graph their answers on the handout provided. A classroom participation grade will be given on this group work. The groups are….

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Try This: Example 1B Graphing the equation and tell whether it is linear. B. y = x2 x x2 y (x, y) –2 –1 1 2 (–2)2 4 (–2, 4) (–1)2 1 (–1, 1) (0)2 (0, 0) (1)2 1 (1, 1) (2)2 4 (2, 4)

Try This: Example 1B Continued Course 3 11-1 Graphing Linear Equations Try This: Example 1B Continued The equation y = x2 is not a linear equation because its graph is not a straight line.

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Try This: Example 1C Graph the equation and tell whether it is linear. C. y = x x y (x, y) –8 –6 4 8 –8 (–8, –8) –6 (–6, –6) (0, 0) 4 (4, 4) 8 (8, 8)

Try This: Example 1C Continued Course 3 11-1 Graphing Linear Equations Try This: Example 1C Continued The equation y = x is a linear equation because the points form a straight line. Each time the value of x increases by 1, the value of y increases by 1.

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Try This: Example 1D Graph the equation and tell whether it is linear. D. y = 7 x 7 y (x, y) –8 –4 4 8 7 7 (–8, 7) 7 7 (–4, 7) 7 7 (0, 7) 7 7 (4, 7) 7 7 (8, 7) For any value of x, y = 7.

Try This: Example 1D Continued Course 3 11-1 Graphing Linear Equations Try This: Example 1D Continued The equation y = 7 is a linear equation because the points form a straight line. As the value of x increases, the value of y has a constant change of 0.

Additional Example 2: Sports Application Course 3 11-1 Graphing Linear Equations Additional Example 2: Sports Application A lift on a ski slope rises according to the equation a = 130t + 6250, where a is the altitude in feet and t is the number of minutes that a skier has been on the lift. Five friends are on the lift. What is the altitude of each person if they have been on the ski lift for the times listed in the table? Draw a graph that represents the relationship between the time on the lift and the altitude.

Additional Example 2 Continued Course 3 11-1 Graphing Linear Equations Additional Example 2 Continued

Graphing Linear Equations Course 3 11-1 Graphing Linear Equations Additional Example 2 Continued (Chart might look like this…)

Additional Example 2 Continued Course 3 11-1 Graphing Linear Equations Additional Example 2 Continued The altitudes are: Anna, 6770 feet; Tracy, 6640 feet; Kwani, 6510 feet; Tony, 6445 feet; George, 6380 feet. This is a linear equation because when t increases by 1 unit, a increases by 130 units. Note that a skier with 0 time on the lift implies that the bottom of the lift is at an altitude of 6250 feet. Notice the broken line!

Summary of Steps for Graphing Equations 1. Choose a value for x 2. Substitute the x-value into the equation & find the corresponding y-value 3. Form an ordered pair with the x and y-value 4. Graph the ordered pair 5. Repeat the process to achieve a minimum of 3 points

Graphing Linear Equations Insert Lesson Title Here Course 3 11-1 Graphing Linear Equations Insert Lesson Title Here Lesson Quiz Graph each equation and tell whether it is linear. 1. y = 3x – 1 2. y = x 3. y = x2 – 3 yes 14 yes no

Homework for 12-10-07! Answer the Essential Question on your homework paper for tonight Homework workbook section 11-1