Lesson 6
Representations of Linear Relationships
Let’s write equations from real situations.
Problem 1
Create a graph that shows three linear relationships with different \(y\)-intercepts using the following slopes, and write an equation for each line.
Slopes:
- \(\frac15\)
- \(\frac35\)
- \(\frac65\)
![Graph of quadrant 1.](https://staging-cms-im.s3.amazonaws.com/azTun3TyHPPGrYTw6TAKivQW?response-content-disposition=inline%3B%20filename%3D%228-8.3.B7.PP.1stquadxy.png%22%3B%20filename%2A%3DUTF-8%27%278-8.3.B7.PP.1stquadxy.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T141519Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c5355f0f0e77fa7463c9553d7b906ba384ff6a3babbb09e954c9b2f001259468)
Problem 2
The graph shows the height in inches, \(h\), of a bamboo plant \(t\) months after it has been planted.
![Graph, t, months, h, inches.](https://staging-cms-im.s3.amazonaws.com/udpycGytSEqwjDzzLTJXqKRA?response-content-disposition=inline%3B%20filename%3D%228-8.3.B.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.3.B.PP.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T141519Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0ad869e632347ae1f208cad2094597631dd2542aa31210099520f18b29b17d75)
- Write an equation that describes the relationship between \(h\) and \(t\).
- After how many months will the bamboo plant be 66 inches tall? Explain or show your reasoning.
Problem 3
Here are recipes for two different banana cakes. Information for the first recipe is shown in the table.
sugar (cups) | flour (cups) |
---|---|
\(\frac12\) | \(\frac34\) |
\(2\frac12\) | \(3\frac34\) |
3 | \(4\frac12\) |
The relationship between cups of flour \(y\) and cups of sugar \(x\) in the second recipe is \(y=\frac74x\)
- If you used 4 cups of sugar, how much flour does each recipe need?
- What is the constant of proportionality for each situation and what does it mean?
Problem 4
Show that the two figures are similar by identifying a sequence of translations, rotations, reflections, and dilations that takes the larger figure to the smaller one.
![figure ABCD and figure EFGH on a grid.](https://staging-cms-im.s3.amazonaws.com/3EtfbbrKxN68Z5KAzaexR6Bh?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T141519Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7d729cc745cf3cbf0d1c60bd7ec03fa4ea7c5ebd70762ca1bcd0bab82849824c)