lesson 2: homework
notes or additional instructions based on whole-class discussion of homework assignment:

1. what are important graphical features of proportional relationships? circle all correct statements:
a. the graph is a straight line.
b. the line has a y-intercept of 0.
c. the line passes through the origin.
d. the line has an x-intercept of 0.
e. the line passes through the point (0,0).

use the graph to answer questions 2–5.

2. which lines represent proportional relationships?
explain how you know.

3. find the rate of change for each line.
a:
b:
c:
d:
e:

4. find the constant of proportionality for each proportional relationship.

5. write a function rule for each proportional relation

(graph description: a coordinate plane with lines a, b, c, d, e. line a is steep, passing near (0,0) and (1,4+). line b is shallow, starting at (0,4). line c starts at (0,2). lines d and e pass through the origin (0,0), with d steeper than e.)

Question

lesson 2: homework
notes or additional instructions based on whole-class discussion of homework assignment:

1. what are important graphical features of proportional relationships? circle all correct statements:
   a. the graph is a straight line.
   b. the line has a y-intercept of 0.
   c. the line passes through the origin.
   d. the line has an x-intercept of 0.
   e. the line passes through the point (0,0).

use the graph to answer questions 2–5.

2. which lines represent proportional relationships?
   explain how you know.

3. find the rate of change for each line.
   a:
   b:
   c:
   d:
   e:

4. find the constant of proportionality for each proportional relationship.

5. write a function rule for each proportional relation

(graph description: a coordinate plane with lines a, b, c, d, e. line a is steep, passing near (0,0) and (1,4+). line b is shallow, starting at (0,4). line c starts at (0,2). lines d and e pass through the origin (0,0), with d steeper than e.)

Accepted Answer

### Question 1
# Explanation:
To determine the correct statements about the graphical features of proportional relationships, we recall the definition of a proportional relationship. A proportional relationship is of the form $ y = kx $, where $ k $ is the constant of proportionality.

## Step 1: Analyze statement a
A proportional relationship is a linear function, so its graph is a straight line. So statement a is correct.

## Step 2: Analyze statement b
For $ y = kx $, when $ x = 0 $, $ y = 0 $. So the $ y $-intercept is $ 0 $. Statement b is correct.

## Step 3: Analyze statement c
Since when $ x = 0 $, $ y = 0 $, the line passes through the origin $(0,0)$. Statement c is correct.

## Step 4: Analyze statement d
The $ x $-intercept is found by setting $ y = 0 $. For $ y = kx $, $ 0=kx $ implies $ x = 0 $ (assuming $ k
eq0 $). So the $ x $-intercept is $ 0 $. Statement d is correct.

## Step 5: Analyze statement e
As shown in steps 2 and 3, the line passes through $(0,0)$. Statement e is correct.

# Answer:
a. The graph is a straight line.
b. The line has a $ y $-intercept of 0.
c. The line passes through the origin.
d. The line has an $ x $-intercept of 0.
e. The line passes through the point $(0,0)$.

### Question 2
# Explanation:
A proportional relationship has the form $ y = kx $, which means the line must pass through the origin $(0,0)$ (since when $ x = 0 $, $ y = 0 $) and be a straight line.

## Step 1: Examine each line
- Line A: Passes through the origin? From the graph, Line A starts at the origin? Wait, looking at the graph, Line A: Let's check the intercept. Wait, the graph: Line A, B, C, D, E. Wait, Line B has a $ y $-intercept of 4, Line C has a $ y $-intercept of 2, Line A: Wait, maybe I misread. Wait, the origin is $(0,0)$. Lines that pass through $(0,0)$ are A, D, E? Wait, no, let's look again. Wait, the graph: Line A: when $ x = 0 $, $ y = 0 $? Wait, the grid: the origin is at (0,0). Line A: starts at (0,0)? Wait, Line B: at $ x = 0 $, $ y = 4 $. Line C: at $ x = 0 $, $ y = 2 $. Line A: at $ x = 0 $, $ y = 0 $? Wait, the graph shows Line A, D, E passing through (0,0)? Wait, Line A: from (0,0) going up steeply. Line D: from (0,0) with moderate slope. Line E: from (0,0) with shallow slope. Line B: $ y $-intercept 4, Line C: $ y $-intercept 2. So proportional relationships are lines that pass through the origin (since $ y = kx $ has $ y $-intercept 0). So Lines A, D, E pass through the origin, so they represent proportional relationships. Lines B and C have non - zero $ y $-intercepts, so they are not proportional.

# Answer:
Lines A, D, and E represent proportional relationships. We know this because a proportional relationship has the equation $ y=kx $, whose graph is a straight line passing through the origin $(0,0)$. Lines A, D, and E pass through the origin, while Lines B and C have non - zero $ y $-intercepts (they do not pass through the origin) and thus do not represent proportional relationships.

### Question 3
To find the rate of change (slope) of a line, we use the formula for slope $ m=\frac{y_2 - y_1}{x_2 - x_1} $. For proportional relationships (passing through the origin $(0,0)$), we can use a point $(x,y)$ on the line and calculate $ \frac{y}{x} $.

## Line A
## Step 1: Identify a point on Line A
Looking at the graph, when $ x = 1 $, let's see the $ y $-value. From the graph, Line A at $ x = 1 $, $ y = 4 $ (approx? Wait, the grid: let's assume the grid has 1 - unit squares. Wait, Line A: from (0,0) to (1,4)? Wait, no, maybe at $ x = 1 $, $ y = 4 $? Wait, the graph: the $ y $-axis has marks at 1,2,3,4,5. Line A: when $ x = 1 $, $ y = 4 $? Wait, no, maybe I made a mistake. Wait, the slope formula: for a line through $(0,0)$ and $(x,y)$, slope $ m=\frac{y}{x} $.

Wait, let's take two points. For Line A: Let's say it passes through (0,0) and (1,4) (from the graph). Then slope $ m=\frac{4 - 0}{1 - 0}=4 $.

## Line B
Line B is not a proportional relationship (has $ y $-intercept 4), but if we still calculate the slope. Let's take two points. At $ x = 0 $, $ y = 4 $; at $ x = 4 $, $ y = 5 $ (approx). Then slope $ m=\frac{5 - 4}{4 - 0}=\frac{1}{4}=0.25 $. But since it's not proportional, but the question asks for rate of change (slope) for each line.

## Line C
Line C is not proportional ( $ y $-intercept 2). Take two points: (0,2) and (4,4). Slope $ m=\frac{4 - 2}{4 - 0}=\frac{2}{4}=0.5 $.

## Line D
Line D is proportional (passes through (0,0)). Take a point, say (4,2). Then slope $ m=\frac{2 - 0}{4 - 0}=\frac{2}{4}=0.5 $. Wait, no, maybe (2,1): slope $ \frac{1 - 0}{2 - 0}=0.5 $.

## Line E
Line E is proportional (passes through (0,0)). Take a point, say (6,1). Then slope $ m=\frac{1 - 0}{6 - 0}=\frac{1}{6}\approx0.1667 $.

# Answer:
- A: Let's assume a point (1,4) on Line A, slope $=\frac{4 - 0}{1 - 0}=4$
- B: Using points (0,4) and (4,5), slope $=\frac{5 - 4}{4 - 0}=\frac{1}{4}=0.25$
- C: Using points (0,2) and (4,4), slope $=\frac{4 - 2}{4 - 0}=\frac{2}{4}=0.5$
- D: Using points (0,0) and (4,2), slope $=\frac{2 - 0}{4 - 0}=0.5$
- E: Using points (0,0) and (6,1), slope $=\frac{1 - 0}{6 - 0}=\frac{1}{6}\approx0.167$

### Question 4
The constant of proportionality $ k $ for a proportional relationship $ y = kx $ is equal to the slope of the line (since $ k=\frac{y}{x} $ for a point $(x,y)$ on the line).

## Line A
From Question 3, the slope of Line A is 4, so the constant of proportionality $ k = 4 $.

## Line D
The slope of Line D is 0.5, so the constant of proportionality $ k = 0.5=\frac{1}{2} $.

## Line E
The slope of Line E is $\frac{1}{6}$, so the constant of proportionality $ k=\frac{1}{6}$.

# Answer:
- A: $ k = 4 $
- D: $ k = 0.5 $ (or $ \frac{1}{2} $)
- E: $ k=\frac{1}{6} $

### Question 5
A function rule for a proportional relationship is of the form $ y = kx $, where $ k $ is the constant of proportionality (slope) found in Question 4.

## Line A
Since $ k = 4 $, the function rule is $ y = 4x $.

## Line D
Since $ k = 0.5=\frac{1}{2} $, the function rule is $ y = 0.5x $ (or $ y=\frac{1}{2}x $).

## Line E
Since $ k=\frac{1}{6} $, the function rule is $ y=\frac{1}{6}x $.

# Answer:
- A: $ y = 4x $
- D: $ y = 0.5x $ (or $ y=\frac{1}{2}x $)
- E: $ y=\frac{1}{6}x $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step 1: Analyze statement a

Step 2: Analyze statement b

Step 3: Analyze statement c

Step 4: Analyze statement d

Step 5: Analyze statement e

Step 1: Examine each line

Answer:

Question 2