Chapter 4 Graphing Linear Equations and Functions 4.1 Coordinates and Scatter plots on the calculator: On the graph paper below please put the following items: x and y axis, origin,quadrant numbering system, and graph an ordered pair in each quadrant using (x,y). Who discovered the coordinate system? The graph of a scatter plot can have three kinds of correlation. Name them, and describe that each one looks like.( Or draw a sketch of each one) What is an outliers within a set of data? Now turn to page 205 in your text book and construct a scatter plot with your graphing calculator using the data in the snowmobile problem at the top of the page. Remember there are several things you have to do: 1) Go to list and input your data 2) Go to Stat Plot and set up your information for a scatter plot 3) Go to the window and set both the x and y ranges 4) Press Graph Pg. 209 in your textbook will help you set up your scatterplot if you have forgotten how to do it from last year even with the directions given above.
Quick Review: 2 Name this form of a linear function 2x +4y=12 Name this form of a linear function y= 2x 5 Can you describe what a solution, to either one of the above equations, looks like? Can you explain why these equations have 2 different variables? What would you put the solutions, to either one of the equations above,on? 4.2 Graphing Linear Equations: Linear Equations in one variable (3,3), ( 3.-2), ( 3,-1), ( 3,0), ( 3,1), (3, 2) Graph these ordered pairs here! AND write the coordinate next to each point you graph. Label with x and Y axes 1) What is unique about these ordered pairs? 2)The graph of these ordered pairs is what kind of line? 3) What is the equation of this line? 4) How would you describe this line? 5) This line is parallel to which axis? 6) Where do you find the graph of an ordered pair when the x coordinate is zero? 7) Where do you find the graph of an ordered pair when the y coordinate is zero?
3 ( 2, -3), ( 4, -3), ( 0, -3), ( 1, -3), ( -2,-3), (-3, -3) Graph these ordered pairs here! AND write the coordinate next to each point you graph. Label with x and Y axes 1) What is unique about these ordered pairs? 2)The graph of these ordered pairs is what kind of line? 3) What is the equation of this line? 4) How would you describe this line? 5) This line is parallel to which axis? Now graph the following equations of a line on the graph provided, and give me the coordinates of the point of intersection. x = -4, y = 3 x=0, y=1 Point of intersection Point of intersection Describe the graph of the line x= 4 Describe the graph of the line y = 1
4.2 Graphing Linear Equations in Two Variables: One way to find solutions to a linear equation so you can graph it is to make a table of values. You can use it from either standard form or slope-intercept form. You can select values for x. 4 Y = 2x + 1 4x + 2y =1 X Y X Y You can check to see if an ordered pair is a solution of a linear equation by substitution the X and Y coordinates in the equation. Then check to see if the equality is true. Which of the points is on the graph of x + 3y =6? see! (1,2) or ( -2, 8 ) Check and 3 Remember the graph of an equation is a visual model of the equation. a) Sketch the graph of y = x and y = x on the same coordinate plane. What do you notice about the two graphs? Describe the direction of the graph of each line.
b) Complete the table of values for Y = 2x +5 5 X -3 2 1 0 1 2 3 Y 4.3 Quick graphs Using intercepts Intercepts are? What is unique about the ordered pair of each intercept? Name the two intercepts for a linear equation. and How do you find the x-intercept of the line 2x y =4? Show your work.!!!!!!! How do you find the y-intercept of the line 2x y =4? Show your work!!!!! Is 8 the x-intercept or the y-intercept of the line y = x + 4? Prove it. Is the y-intercept of the line y = 6x 4 equal to 6 or 4? Justify your answer with work.
4.4 Slope of the line The slope of the line is a ratio of the rise of the line. the run of the line The formula for slope of a line is : 6 What do the x 1, x 2, y 1, y 2 mean There are four types of slope: 1) Find the slope of ( 2, 0) and ( 3, 1) 2) Find the slope of ( -1, 2) and ( 2,2) Describe the slope Describe the slope 3) Find the slope of ( 0, 0 ) and ( 1, 1) 4) Find the slope of ( 2, 1) and ( 2, 4) Describe the slope Describe the slope
7 Slope is represented by the letter m. What word do you think it stands for? (Hint: It is French.) Turn to page 229 and look at example 6. A rate of change is also called slope. Find the constant rate of change for that example without LOOKING at the answer. No peeking!!!! On page 231 find the slope of #47 and #48. Then check with your neighbor to see if they got the same answer but with different ordered pairs than you did. 4.5 Direct Variation: Two variable quantities that have the same rate of, regardless of the values of the variables, have a variation. For example if you get $5 per hour, then your total varies directly with the number of you work. Check it out below. Total pay (y) $30 $50 $100 $350 Hours worked(x) 6 10 20 70 Rate of pay?(k) A MODEL for all Direct Variations: y varies directly as x = y x This variation is always equal to a k constant. y x = k or y = kx. So in our example above Your total pay varies directly as your number of hours worked. (Hint: The more hours you work the more money you get paid.) But the rate you are paid per hour NEVER CHANGES. ( Unless you get a raise.) So Total Pay #of hours worked = K K is the constant value ( your hourly rate of pay) that does not change as the y and x values change.
8 Decide if these statements are direct variations. a) You are riding your bike at an average speed of 12 miles per hour. The number of miles you ride, d, during, t, hours is given by d = 12t. b) The time, t, spent waiting in line for the Super Looper roller coaster, and the number of people, n, in the line are related by the equation t = Kn. c) If y varies directly as x and y = 15 and x = 2 find the constant,k. Now we will apply this to the Slope of a line. Where the constant, k, is of the line no matter two ordered pairs you from the line. Y= the change in y 2 y 1 x = the change in x 2 x 1 The variables x and y vary directly. When x = 15 and y = 105 write an equation that relates x and y, Find the value of the constant. Now find the value of y when x =12. y= Graph the equation. Find the constant of variation and the slope of the line for y = x Turn to page 238 and look at the Violin Family problem.
9 4.6 Quick Graphs Using Slope-Intercept Form: To make a quick graph of a linear equation: x + 3y = 6 First Next pluck out and Now graph and then use over to graph the line. Now! Do it again with the two examples below. 3x 6y = 9 x 2y = 3
Give the slope and y-intercept for each of the following equations EQUATION SLOPE Y- INTERCEPT 1) y = 2x + 3 10 2) y = 2x 3) y = 2x 1 Now graph all three on the graph below What is unique about all three graphs? What does the slope have to do with this? Summarize what you have learned about the slope and the graphs of these lines. EQUATION SLOPE Y- INTERCEPT 1) y = 1 2 x + 3 2) y = 1 2 x 1 3) y = 1 2 x + 2 Now graph all three on the graph below What is unique about all three graphs? What does the slope have to do with this? Summarize what you have learned about the slope and the graphs of these lines.
11 4.7 Solving Linear Equations Using Graphs: One variable equation: Two variable equation 0 = 2x 8, solve for x y = 2x 8 Now substitute the value for x into the Two variable equation and what are you going to solve for? What intercept are we using the one variable equation to solve for? Forms of linear equations: a) Slope intercept form b) Y intercept form c) Solve for the x-intercept. Write the slope- intercept form of the equation as it would look after you ve replaced the y with a zero. This should look like the One Variable Equation at the top of the page. TRUE or FALSE? The solution of 0 = 9x - 36 is the x-intercept of y = 9x 36? Explain your reasoning and show your work? Now try another one. 2x 5 = 1 Solve for x. Don t forget to check your solution back in the equation. Write the equation in the form ax + b = c _ x + _ = _ Replace the zero with a Y. Y = _ x + _ Now check the function by graphing it and checking the intercepts!!!
12 Now do it again with 4 3 x + 3 = 6 x and follow the steps from the last example. 3 Don t forget to graph and check those intercepts, 4.8 Functions and Relations: A relation is defined as a set of pairs (x,y). A function is a relation in which NO two ordered pairs have the same x-value. Input Output Input Output 1 5 1 2 2 7 2 4 3 9 3 5 4 4 Write the set of ordered pairs that is shown by each Input-Output above. Check the ordered pairs to see if the criteria for the relation or the function are met? Graph each set of ordered pairs on the graphs on the next page.
13 Now we will use the drop -line test to see if the graphs are a relation or a function. Get out your small ruler out to use as a vertical line tester. Drag your ruler across each graph, keeping it parallel to the Y axis. The question is does it intersect only one point or more than one point at any one time?????? Vertical line test: If the vertical line intersects with only point it is a If the vertical line intersects with more than point it is not a but it is a. Now copy the graphs, below, in the same order I put them on the board. Function or Relation? X-values are called the of the function( input). Y-values are called the of the function (output).