Lesson 8
Using Graphs to Compare Relationships
Problem 1
The graphs below show some data from a coffee shop menu. One of the graphs shows cost (in dollars) vs. drink volume (in ounces), and one of the graphs shows calories vs. drink volume (in ounces).
__________________ vs volume
![Graph of points on coordinate plane, origin O.](https://cms-im.s3.amazonaws.com/yLRkVa31GvVxPFPMmnjYuFzZ?response-content-disposition=inline%3B%20filename%3D%227-7.2.E.PP.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.E.PP.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181531Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9fea754db88dbb70b8126c39e8d20bc2744ac5d405ced2676db6e6696721256a)
_____________________ vs volume
![Graph of points on coordinate plane, origin O.](https://cms-im.s3.amazonaws.com/fsenBttzN9Nqqr5GfA3JH4CW?response-content-disposition=inline%3B%20filename%3D%227-7.2.E.PP.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.E.PP.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=936fe5f8e8763cb1d7d76afea387d6994f8f9dc156c36b73ee96f2247a2864c6)
- Which graph is which? Give them the correct titles.
- Which quantities appear to be in a proportional relationship? Explain how you know.
- For the proportional relationship, find the constant of proportionality. What does that number mean?
Solution
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Problem 2
Lin and Andre biked home from school at a steady pace. Lin biked 1.5 km and it took her 5 minutes. Andre biked 2 km and it took him 8 minutes.
- Draw a graph with two lines that represent the bike rides of Lin and Andre.
- For each line, highlight the point with coordinates \((1,k)\) and find \(k\).
- Who was biking faster?
Solution
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Problem 3
Match each equation to its graph.
- \(y = 2x\)
- \(y = \frac45 x\)
- \(y = \frac14 x\)
- \(y = \frac23 x\)
- \(y = \frac43 x\)
- \(y = \frac32 x\)
1
![Graph of a linear function, origin O.](https://cms-im.s3.amazonaws.com/r3ba1ZWBYSGKQQ2SD7jZAczQ?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.07.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.07.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=66cc1fc9c8e67464528e66c9ceb20bdf64450c7c19191cf80fb690e70df72f88)
2
![Graph of a linear function, origin O.](https://cms-im.s3.amazonaws.com/sdqCv7iEZSnPDh56Bix9GpTr?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c50195bacc114cf66ec17ea13b0deb2e760f15c93dae842a437b35fe998ba958)
3
![Graph of a linear function, origin \(O\).](https://cms-im.s3.amazonaws.com/kNzh7TJAb3gK64G13VnqinfF?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.09.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.09.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=170369d34dc9e9ea67eafeaf0de4cc6ae9a9cc35ba67d98ce9a58462a2ecadc0)
4
![Graph of a linear function, origin O.](https://cms-im.s3.amazonaws.com/4MqnAajWEeg4rVPrwHE3Jczm?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3e76c379a51868bc318a12025bd8c3ec7b3a6e11b1174491bb404ce56c86848e)
5
![Graph of a linear function, origin O.](https://cms-im.s3.amazonaws.com/XEbpdJ3eWpn7uiDrPtzRVfYg?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f28ee05d55efe5687851f295a28d6b5c30ea73a283901ffc9b9b0039be85c273)
6
![Graph of a linear function, origin O.](https://cms-im.s3.amazonaws.com/esdeNuoKCqSTtgV9DyCWDosR?response-content-disposition=inline%3B%20filename%3D%227-7.2.F.PP.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.F.PP.Image.12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181532Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f3428c817fd57a25b708114c6d221769bc0feec430f4ea3d0708d11cd64572f1)
Solution
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