Chapter 4 Linear Relations

4.1 – Writing equations to describe patterns 4.2 – Linear relations Chapter 4

Linear relations A linear relation is a relation that has a graph that is a straight line. They help us to define patterns in a way that we can use to figure out what will happen. Follow along on the handout as we go through the problems. Make sure that you answer all of the questions.

handout Sketch: How would we describe what’s happening in words? 4 Equation: How would you find a value for p if you knew the value for b? b = 3p + 1 7 10 13 Review of Substitution: If you know that there are 9 plots, let p = 9.  b = 3(9) + 1 = 28 boards

handout Try examples 2 and 3 on your own. Make sure that you answer all the questions to the fullest of your abilities, as it will be a summative assessment.

example When a scuba diver goes under water, the weight of the water exerts pressure on the diver. What patterns do you see in the table and in the graph? What do these patterns tell you about the relationship between depth and water pressure?

example Equation:

Linear relations Linear Relation: When the graph of the relation is a straight line, we have a linear relation. In a linear relation, a constant change in one quantity produces a constant change in the related quantity.

Example x y –3 –2 –1 1 2 3 c) We move one unit right, and three units down. In the table, when x increases by 1, y decreases by 3. 15 12 9 6 d) The relation is linear because the change in y is constant, and because the graph is a straight line. 3 –3

example The student council is planning to hold a dance. The profit in dollars is 4 times the number of students who attend, minus $200 for the cost of the music. Write an equation that relates the profit to the number of students who attend. Create a table of values for this relation. Graph the data in the table. Does it make sense to join the points? Explain. How many students have to attend to make a profit? Let P be profit, and s be the number of students. P = 4s – 200 c) d) There is a profit if P > 0. 4s – 200 > 0 4s > 200 s > 50 There needs to be at least 51 students for a profit to be made. b) s P –200 50 100 200

Pg. 159-162, # 5, 7, 8, 10, 13, 16, 18 pg. 170-173. # 5, 7, 8, 10, 13, 14, 16, 18 Independent Practice

4.3 – Another form of the equation for a linear relation Chapter 4

Y=mx+b GeoGebra Applet We will often see linear equations in the form y = mx + b, and this form tells us a lot about the way that a linear relation will look like when it is graphed. When you have an equation and x and y are on the same side of the equal sign then we need to solve for y to put it into this form. In linear relations, neither x or y will have an exponent.

If x or y is missing from the relation

example Graph: a) y + 2 = 0 b) 2x = 5 y y a) b) y + 2 = 0  y = –2

example For the equation 3x – 2y = 6 Make a table of values for x = –4, 0, and 4 Graph the equation. y Start by solving for y: 3x – 2y = 6 2y = 3x – 6 y = (3x – 6)/2 b) x x y –4 –9 –3 4 3

Matching game Match the graphs to their given equations as quickly as you can. What were some of the strategies that you used?

pg. 178-180. # 4, 7, 8, 12, 14, 18, 19, 21 Independent Practice

4.4 – matching equations and graphs Chapter 4

example

Example Which graph on this grid has the equation y = 3x – 4? Justify your answer. We know, from the form y = mx + b, that b is the y-intercept. So, the y-intercept (or where the graph passes through y) must be –4. The equation is graph III. What are some other ways that we could determine which graph it is?

Try it

What were your estimates for Part 1? continued What were your estimates for Part 1? For Part 2? For Part 3? What are some strategies that you used in your predictions?

example Jenna borrows money from her parents for a school trip. She repays the loan by making regular weekly payments. The graph shows how the money is repaid over time. The data are discrete because payments are made every week. How much money did Jenna originally borrow? How much money does she still owe after 3 weeks? How many weeks will it take Jenna to repay one-half of the money she borrowed? c) Half of 200 is 100. We can see from the graph that at $100, that it is between 4 and 6 weeks.  It will take her 5 weeks to pay half of it. How much money did she owed at Time = 0?  From the graph we can see that she owed $200. b) At three weeks, we can see she owes half way between 120 and 160 dollars. So, she must owe 140 dollars.

Try it What’s the pattern that we see on the graph? What happens every time y increases by 1? As y increases by 1, x increases by 3. So, when y decreases by 1, x decreases by 3. What’s the value of x at y = 4? it’s x = –3. So, at y = 3, x = –3 – 3 = –6 b) We can use the graph to estimate. We can see that it is more than 6 but less than 7. It looks approximately two-thirds of the way to 7, so we can say that when x = 5, y = 6.666…

pg. 188-190. # 5, 6, 7, 9, 11, 13 pg. 196-198 # 4, 6, 8, 11, 13, 15. Independent Practice

handout Follow the instructions on the handout, and answer all of the questions to the best of your ability. This will be a summative assessment.

Graphing on your calculator To graph: Y= Enter your Equation GRAPH Change your window settings: WINDOW To see a Table of Values: You can scroll through the values using the up and down arrows. 2nd GRAPH To find an y-value, when you know an x-value: TRACE X-VALUE NUMBER ENTER

Using your calculator to graph a table of values To create our Table of Values: Put your x-values underneath L1 Put your y-values underneath L2 (use the arrow keys to navigate). STAT ENTER To find the equation: On your screen it will give you the variables for the equation y = mx + b. Write down your equation. STAT CALC 4 ENTER To graph: x y 9 8 11 7 14 6 17 5 20 Y= Enter your Equation GRAPH You may need to change your window settings to see enough of the graph.