b Recall that each participant's head start is their y-intercept, and their rate is the slope.
C
c Use the general form for a linear relationship, y=mx+b, to summarize the information provided by the graph.
D
d How can you create a unit rate from the slope of his line?
E
e Set the equations equal to one another.
A
aDependent Variable: Distance
Independent Variable: Time
B
bWinner: Mark
C
cBarbara: y=3/2x+3
Mark: y=4x
D
d 2.5 m/s.
E
e After 2 seconds, at 6 meters.
Practice makes perfect
a The independent variable refers to the input of a function, while the dependent variable is the output of the function. The value of the dependent variable depends on the value of the independent variable. In our case, we have the following.
Independent (x):& Time in seconds
Dependent (y):& Distance in meters
b Let's consider the information for each participant one at the time.
Barbara
Barbara began the race with a 3 meter head start. Previously, we learned that the headstart of different participants correspond to the y-intercept of their line. We also know that she went 3 meters every 2 seconds. We can use this to create a second point and draw our line.
Mark
Mark began at the starting line, so we know that his y-intercept must correspond to the origin. We also know that he finished the entire race, 20 meters, in 5 seconds. This means that a second point on the line is going to be (5,20).
Carlos
For Carlos, we were already given the equation of his line, y= 52x+1. This tell us that his y-intercept is 1 and he received a 1 meter head start. We also know that he rode 5 meters every 2 seconds.
Combined Graphs
We can combine all three of these lines on the same coordinate plane to compare the three participants.
c Let's look at the information provided by the graphs we created in Part B so that we can write an equation for each racer.
In the equation, m is the slope and b is the y-intercept. We already know that Barbara's y-intercept is 3. We can find the slope of the line by thinking about what the changes in vertical and horizontal distance are for this situation.
Δ y/Δ x=Change in distance/Change in time ⇒ m=3/2
The final equation for Barbara's line is as follows.
y= 32x+3
Mark
Using the same process, we can find the equation for Mark's line. We know that his y-intercept is 0. Let's also find the slope.
Δ y/Δ x=Change in distance/Change in time ⇒ m=20/5=4
Therefore, the final equation for Mark is the following.
y=4x
d To find the speed at which Carlos rode, we can use the following formula.
d=r* t
In this formula d is distance, r is rate, and t is the time spent traveling. Based on the slope of his equation, we know that he pedaled 5 meters every 2 seconds. Let's use this information to solve for his speed.
Now that we have found the time at which they were equal distances from the starting line, we can confirm the distance when that occurred.
Carlos:& 52* 2+1=5+1= 6 [0.8em]
Barbara:& 32* 2+3=3+3= 6
They were both at the 6 meter mark after 2 seconds.
