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Algebra 1 Ch 4.2 – Graphing Linear Equations Objective Students will graph linear equations using a table. Students will graph horizontal and vertical lines Vocabulary A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. The solution to linear equations are ordered pairs which makes the equation true. The graph of an equation in x and y is the set of all points (x, y) that are solutions of the equation. Example #1 This is an example of a linear equation y=x+8 All linear equations are functions. That is the value of y (output) is determined by the value of x (input) The linear equation in two variables can also be called the rule. In this case whatever x is plus 8 will give you the value of y Graphing Since the results of a linear equations can be expressed as ordered pairs, the linear equation can be graphed. When a linear equation is graphed, all points on the line represent the solution set of the linear equation. There are a number of ways to find the solution to a linear equation…for today’s lesson we will look at creating a table of the solutions… Tables To create a table of solutions to a linear equation do the following: 1. 2. 3. 4. Choose a minimum of 3 values for x Substitute the values of x into the linear equation Simplify to find the value of y Write the solutions as ordered pairs Let’s look at an example… Example #2 1. Choose a minimum of 3 values for x y = 2x – 1 x 0 y = 2x – 1 y = 2(0) – 1 y -1 (x, y) (0, - 1) 1 y = 2(1) – 1 1 (1, 1) 2 y = 2(2) – 1 3 (2, 3) 2. Substitute the value of x into the equation 3. Simplify to determine the value of y 4. Write as an ordered pair Comments How do you choose the value of x? You can use any number for the value of x…however, you can make your life easy by choosing -1, 0, and 1, because you can do mental math and they are easy to graph. You choose a minimum of 3 numbers because when graphed the expected result is a straight line. If you don’t get the straight line, then you have to go back and check your calculations… Graphing Use the ordered pair from the table to graph the linear equation. Again…when graphing the result should be a straight line… Any point (ordered pair) on that line will be a solution to the linear equation… y (2,3) x (1,1) (0, -1) (x, y) (0, - 1) (1, 1) (2, 3) More Comments… Once you get the hang of it…this concept is really easy…. However, like everything else to become proficient you will need to practice… It’s ok if you make a mistake…you can fix it… The goal here is to keep an open mind and try… Let’s look as some special equations that produce horizontal and vertical lines… Linear Equations All linear equations can be written in the form: Ax + By = C This form is called the standard form of an equation. At this level you are required to know and be able to manipulate this form of an equation Standard Form Ax + By = C In the standard form of an equation: A is the coefficient of x B is the coefficient of y C represents the constant We talked about coefficients and constants in a previous lesson Example #3 The equation 3x – 4y = 12 is an example of an equation written in standard form. As we have done in a previous lesson, we can write the equation in function form by transforming the equation as follows: 3x – 4y = 12 -3x Standard Form -3x – 4y = -3x + 12 –4 –4 y=-¾x+3 Function Form Horizontal Lines In the standard form of an equation Ax + By = C, When A=0 the equation reduces to By = C and the graph will be a horizontal line. We often see this illustrated as the equation y = b. In this instance, the equation has no x-value and the y-value is always the same number so that when the y-value is graphed a horizontal line is produced. Example #4 – Horizontal Line Graph the equation y=2 In this instance there is no x-value. All the y-values = 2 To plot this line, starting at 0, go up 2 spaces on the y-axis and draw a horizontal line (as shown in the next slide) Example #4 (Continued) y=2 y y=2 x Comments Notice that when you graph the line, the line is perpendicular to the y-axis. A common error that students make when graphing an equation like y=2 is that they draw the line parallel to the y-axis. That is incorrect! A way to avoid this error is to actually plot the point before you draw the line. Vertical Lines In the standard form of an equation Ax + By = C, When B=0 the equation reduces to Ax = C and the graph will be a vertical line. We often see this illustrated as the equation x=a In this instance, the equation has no y-value and the x-value is always the same number so that when the x-value is graphed a vertical line is produced Example #5 – Vertical Line Graph the equation x = -3 In this instance there is no y-value. All the x-values = -3 To plot this line, starting at 0, go 3 spaces to the left on the x-axis and draw a vertical line (as shown in the next slide) Example #5(Continued) x=-3 y x = -3 x Comments Notice that when you graph the line, the line is perpendicular to the x-axis. A common error that students make when graphing an equation like x=-3 is that they draw the line parallel to the x-axis. That is incorrect! A way to avoid this error is to actually plot the point before you draw the line. Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help… Your Turn 1. 2. 3. 4. Find 3 different ordered pairs that are the solutions to the equation y = 3x – 5 y = -2x – 6 y = ½ (4 – 2x) y = 4( ½ x – 1) Your Turn 5. 6. 7. 8. 9. 10. Rewrite the equation in function form 2x + 3y = 6 5x + 5y = 19 Create a table of values & graph the linear equation y = -x + 4 y= -(3 – x) x=9 y = -1 Your Turn Solutions 1. 2. 3. 4. 5. 6. (-1,-8), (0,-5),(1, -2) (-1,-4),(0,-6),(1,-8) (-1,3),(0,2),(1,1) (-1,-6),(0,-4),(1,-2) y = -2/3x + 2 y = -x + 19/5 7. 8. 9. 10. You should have a table with a minimum of 3 values. When plotting the line the following should be true: Your graph should cross the y-axis at +4 Your graph should cross the y-axis at -3 You should have a vertical line at the point x = 9 You should have a horizontal line at the point y= -1 Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about Graphing Linear Equations Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide… Credit I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…