Unit 4, Lesson 11
Connecting Equations to Graphs (Part 2)
Integrated Math 1
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Here are two graphs that represent situations you have seen in earlier activities.
Graph of a line, origin O. Horizontal axis,almonds, pounds, scale is 0 to 14, by 1’s. Vertical axis, dried figs, pounds, scale is 0 to 10, by 1’s. Line starts at 0 comma 8 point 2 5, passes through 2 comma 7, 5 comma 5, and 11 comma 1.
The first graph represents , which describes the relationship between gallons of water in a tank and time in minutes.
The second graph represents . It describes the relationship between pounds of almonds and figs and the dollar amount Clare spent on them.
Suppose a classmate says, “I am not sure that the graph represents because I don’t see the 6, 9, or 75 on the graph.” How would you show your classmate that the graph indeed represents this equation?
Here are two situations and two equations that represent them.
Situation 1: Mai receives a \$40 bus pass. Each school day, she spends \$2.50 to travel to and from school.
Let be the number of school days since Mai receives a pass and be the balance or dollar amount remaining on the pass.
Situation 2: A student club is raising money by selling popcorn and iced tea. The club is charging \$3 per bag of popcorn and \$1.50 per cup of iced tea, and plans to make \$60.
Let be the bags of popcorn sold and the cups of iced tea sold.
Here are graphs of the equations. On each graph, the coordinates of some points are shown.
Graph of a line, origin O. Horizontal axis, number of school days, scale is 0 to 20, by 2’s. Vertical axis, bus pass balance in dollars, scale is 0 to 50, by 5’s. Line starts at 0 comma 40, passes through 2 comma 35, 10 comma 15, and 16 comma 0.
Graph of a line, origin O. Horizontal axis, bags of popcorn sold, scale is 0 to 22 by 2’s. Vertical axis, cups of ice tea sold, scale is 0 to 60 by 5’s. Line starts at 0 comma 40, passes through 2 comma 36, 10 comma 20, and 20 comma 0.
The 40 in the first equation can be observed on the graph and the -2.50 can be found with a quick calculation. The graph intersects the vertical axis at 40 and the -2.50 is the slope of the line. Every time increases by 1, decreases by 2.50. In other words, with each passing school day, the dollar amount in Mai's bus pass drops by 2.50.
The numbers in the second equation are not as apparent on the graph. The values where the line intersects the vertical and horizontal axes, 40 and 20, are not in the equation. We can, however, reason about where they come from.
What about the slope of the second graph? We can compute it from the graph, but it is not shown in the equation .
Notice that in the first equation, the variable was isolated. Let’s rewrite the second equation and isolate :
Now the numbers in the equation can be more easily related to the graph: The 40 is where the graph intersects the vertical axis and the -2 is the slope. The slope tells us that as increases by 1, falls by 2. In other words, for every additional bag of popcorn sold, the club can sell 2 fewer cups of iced tea.