Lesson 4Tables, Equations, and Graphs of Functions
Let’s connect equations and graphs of functions.
- I can identify graphs that do, and do not, represent functions.
- I can use a graph of a function to find the output for a given input and to find the input(s) for a given output.
4.1 Notice and Wonder: Doubling Back
What do you notice? What do you wonder?
4.2 Equations and Graphs of Functions
The graphs of three functions are shown.
- Match each of these equations to one of the graphs.
- , where is the distance in miles that you would travel in hours if you drove at 60 miles per hour.
- , where is the number of quarters, and is the number of dimes, in a pile of coins worth $12.50.
- , where is the area in square centimeters of a circle with radius centimeters.
- Label each of the axes with the independent and dependent variables and the quantities they represent.
- For each function: What is the output when the input is 1? What does this tell you about the situation? Label the corresponding point on the graph.
- Find two more input-output pairs. What do they tell you about the situation? Label the corresponding points on the graph.
Are you ready for more?
A function inputs fractions between 0 and 1 where and have no common factors, and outputs the fraction . For example, given the input the function outputs , and to the input the function outputs . These two input-output pairs are shown on the graph.
Plot at least 10 more points on the graph of this function. Are most points on the graph above or below a height of ? Of height ?
4.3 Running around a Track
- Kiran was running around the track. The graph shows the time, , he took to run various distances, . The table shows his time in seconds after every three meters.
0 3 6 9 12 15 18 21 24 27 0 1.0 2.0 3.2 3.8 4.6 6.0 6.9 8.09 9.0
- How long did it take Kiran to run 6 meters?
- How far had he gone after 6 seconds?
- Estimate when he had run 19.5 meters.
- Estimate how far he ran in 4 seconds.
- Is Kiran's time a function of the distance he has run? Explain how you know.
Priya is running once around the track. The graph shows her time given how far she is from her starting point.
- What was her farthest distance from her starting point?
- Estimate how long it took her to run around the track.
- Estimate when she was 100 meters from her starting point.
- Estimate how far she was from the starting line after 60 seconds.
- Is Priya's time a function of her distance from her starting point? Explain how you know.
Lesson 4 Summary
Here is the graph showing Noah's run.
The time in seconds since he started running is a function of the distance he has run. The point (18,6) on the graph tells you that the time it takes him to run 18 meters is 6 seconds. The input is 18 and the output is 6.
The graph of a function is all the coordinate pairs, (input, output), plotted in the coordinate plane. By convention, we always put the input first, which means that the inputs are represented on the horizontal axis and the outputs, on the vertical axis.
Lesson 4 Practice Problems
The graph and the table show the high temperatures in a city over a 10-day period.
day 1 2 3 4 5 6 7 8 9 10 temperature (degrees F) 60 61 63 61 62 61 60 65 67 63
What was the high temperature on Day 7?
On which days was the high temperature 61 degrees?
Is the high temperature a function of the day? Explain how you know.
Is the day a function of the high temperature? Explain how you know.
The amount Lin’s sister earns at her part-time job is proportional to the number of hours she works. She earns $9.60 per hour.
Write an equation in the form to describe this situation, where represents the hours she works and represents the dollars she earns.
Is a function of ? Explain how you know.
Write an equation describing as a function of .
Use the equation to complete the table, then graph the line using as the dependent variable.
0 -2 3 0
Solve the system of equations: