Mai Earns $7 Per Hour

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Illustrative Math
Grade 8

Lesson 8: Linear Functions

Permit's investigate linear functions.

Illustrative Math Unit of measurement 8.5, Lesson 8 (printable worksheets)

Lesson eight Summary

Suppose a car is traveling at xxx miles per hour. The relationship betwixt the time in hours and the distance in miles is a proportional relationship. Nosotros tin can represent this human relationship with an equation of the course d = 30t, where distance is a part of time (since each input of time has exactly one output of altitude). Or we could write the equation t = 1/10 d instead, where fourth dimension is a function of altitude (since each input of altitude has exactly one output of time).

More generally, if we represent a linear part with an equation like y = mx + b, then b is the initial value (which is 0 for proportional relationships), and one thousand is the rate of change of the function. If m is positive, the office is increasing. If m is negative, the function is decreasing. If nosotros represent a linear function in a unlike way, say with a graph, we tin can employ what we know near graphs of lines to find the m and b values and, if needed, write an equation.

Lesson 8.one Bigger and Smaller

Diego said that these graphs are ordered from smallest to largest. Mai said they are ordered from largest to smallest. But these are graphs, not numbers! What do yous call up Diego and Mai are thinking?

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Lesson 8.2 Proportional Relationships Define Linear Functions

Jada earns $7 per hour mowing her neighbors' lawns. a. Name the two quantities in this situation that are in a functional relationship. Which did y'all choose to be the independent variable? What is the variable that depends on it? b. Write an equation that represents the function. c. Here is a graph of the function. Label the axes. Label at least two points with input-output pairs.
To convert anxiety to yards, yous multiply the number of feet by 1/3. a. Name the ii quantities in this situation that are in a functional relationship. Which did you lot cull to be the contained variable? What is the variable that depends on information technology? b. Write an equation that represents the function. c. Draw the graph of the role. Characterization at least two points with input-output pairs.

Lesson 8.3 Is it Filling Up or Draining Out?

In that location are 4 tanks of water. The amount of water in gallons, A, in Tank A is given by the part A = 200 + 8t, where t is in minutes. The amount of h2o in gallons, B, in Tank B starts at 400 gallons and is decreasing at five gallons per minute. These functions work when t ≥ 0 and t ≤ 80.

Which tank started out with more than water?
Write an equation representing the relationship between B and t.
One tank is filling up. The other is draining out. Which is which? How tin can you tell?
The amount of h2o in gallons, C, in Tank C is given past the function C = 800 - 7t. Is it filling upwardly or draining out? Tin can you tell simply by looking at the equation? The graph of the function for the amount of h2o in gallons, D, in Tank D at fourth dimension t is shown. Is it filling up or draining out? How practise you know?

Are y'all ready for more than?

Pick a tank that was draining out. How long did it take for that tank to drain? What percent full was the tank when 30% of that time had elapsed? When 70% of the time had elapsed?
What bespeak in the airplane is xxx% of the way from (0,fifteen) to (5,0)? 70% of the way?
What bespeak in the plane is thirty% of the way from (3,five) to (8,6)? 70% of the way?

Lesson 8.iv Which is Growing Faster?

Noah is depositing money in his account every week to save coin. The graph shows the corporeality he has saved as a part of time since he opened his business relationship.
Elena opened an account the same twenty-four hour period as Noah. The amount of money E in her business relationship is given by the function E = 8w + 60, where west is the number of weeks since the account was opened.

Who started out with more money in their account? Explicate how y'all know.
Who is saving money at a faster rate? Explain how you lot know.
How much volition Noah salve over the form of a year if he does not make any withdrawals? How long will it have Elena to save that much?

Lesson eight Practise Problems

Two cars drive on the same highway in the same direction. The graphs testify the altitude, d, of each one as a function of time, t. Which car drives faster? Explicate how yous know.
2 auto services offer to pick y'all up and accept you to your destination. Service A charges 40 cents to option you up and 30 cents for each mile of your trip. Service B charges $1.10 to choice you up and charges c cents for each mile of your trip.
a. Match the services to the Lines l and m.
b. For Service B, is the additional charge per mile greater or less than 30 cents per mile of the trip? Explain your reasoning.
Kiran and Clare like to race each other home from school. They run at the same speed, but Kiran'southward firm is slightly closer to school than Clare's firm. On a graph, their distance from their homes in meters is a function of the time from when they brainstorm the race in seconds.
a. Every bit you lot read the graphs left to right, would the lines become up or downward?
b. What is different about the lines representing Kiran'south run and Clare's run?
c. What is the same about the lines representing Kiran'southward run and Clare's run?
Write an equation for each line.

The Open up Upwards Resources math curriculum is free to download from the Open Resource website and is also available from Illustrative Mathematics.

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