Interpreting the Unit Rate as Slope

Warm Up Give the slope and unit rate.

Graph a proportional relationship and then use the graph to find the slope. Relate the unit rate to the slope. Use a table or a graph to find the unit rate. Use slopes to compare unit rates. Recognize that in a proportional relationship, the constant of proportionality, the unit rate, and the slope of the corresponding graph are the same. Recognize that different relationships have different slopes as well as different unit rates.

Jared is on a 50mile bike ride Jared is on a 50mile bike ride. He wants to finish the ride in under 4 hours. The table shows the distance, d, in miles, that Jared rides his bike in t hours. Find the unit rate and explain whether Jared will meet his goal based on the data.

How do you interpret the unit rate as slope? The ratio of the change in y to the change in x is the unit rate. It is also the ratio of the rise to the run, or the slope. Slope is always rise over run (y-values over x-values). Always read information carefully before deciding on dependent, y-value, and independent, x-value, variables.

The slope of a line that does not go through the origin can be found by its x-intercept and y-intercept. If the x-intercept is a and the y-intercept is b, the line goes through (a, 0) and (0, b). The slope m of the line containing the points is 𝑚= 𝑏 −0 0 −𝑎 =− 𝑏 𝑎

Every 10 seconds an escalator step rises 6 feet Every 10 seconds an escalator step rises 6 feet. Draw a graph of the situation. Find the unit rate of this proportional relationship.

How do you find the slope when you are only given an equation or a table? For an equation such as y = 2.6x, the slope is the unit rate, which is the coefficient of x. For a table, the change in y divided by the change in x is the unit rate, or slope.

Comparing rates, or slopes, is the same as comparing rational numbers Comparing rates, or slopes, is the same as comparing rational numbers. Write each unit rate as a decimal, if necessary.

The equation y = 1.2x represents the rate, in beats per second, that Lee’s heart beats. The graph represents the rate that Nancy’s heart beats. Determine whose heart is beating at a faster rate.

The equation y = 1 9 x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate?

There is a proportional relationship between minutes and dollars per minute, shown on a graph of printing expenses. The graph passes through the point (1, 4.75). What is the slope of the graph? What is the unit rate? Explain.

Two cars start at the same time and travel at different constant rates Two cars start at the same time and travel at different constant rates. A graph for Car A passes through the point (0.5, 27.5), and a graph for Car B passes through (4, 240). Both graphs show distance in miles and time in hours. Which car is traveling faster? Explain.

Exit Ticket Every 4 seconds, a ski lift chair rises 14 feet. Draw a graph of the situation. Then describe the relationship between the height of the chair and the time. Under Plan A, a 2-minute call costs $0.54 and a 4-minute call costs $1.08. Under Plan B, the cost for x minutes is given by y = 0.289x. Which plan is cheaper? Why? The equation y = 13x represents the rate, in gallons per minute, that Tank A at an aquarium fills with water. The table represents the rate that Tank B fills with water. Determine which tank fills faster.