Question

1. which of the following tables represents data that have a proportional relationship?

a

x	y
0	0
1	1.5
3	3.5
6	6.5
8	8.5

b

x	y
0	0
1	1.5
3	4.5
6	9.0
8	12.0

c

x	y
0	0.5
1	1
3	3
6	6
8	8

d

x	y
0	0
1	0.5
3	1.5
6	3
8	5

write an equation of the form ( y = kx ) for the relationship shown in the graph or table.

1. graph: y - axis labeled balloon height (ft), x - axis labeled time (min), with a line passing through (8, 28) and the origin

11.

x	6	10	12
y	45	75	90

(handwritten: ( y = 6x ))

1. use the graph from problem 10. what is the value of ( r ) at the point with coordinates (1, ( r ))? what does this point mean in terms of the proportional relationship shown in the graph?

1. a store sells beans for 80¢ per pound.

a. graph the proportional relationship that gives the cost ( y ) in dollars of buying ( x ) pounds of beans.
b. write an equation of the form ( y = kx ) to represent this relationship. (handwritten: ( y = 1x ))
c. a farmers’ market sells organic locally grown beans for $1.25 per pound. how much more would it cost to buy 3 pounds of beans at the farmers’ market than at the store?

Explanation:

Question 9

Step1: Recall proportional relationship

A proportional relationship has the form \( y = kx \), so \( \frac{y}{x} \) should be constant for all non - zero \( x \), and when \( x = 0 \), \( y = 0 \).

Step2: Check Option A

For \( x = 1,y = 1.5\), \( \frac{y}{x}=1.5\); for \( x = 3,y = 3.5\), \( \frac{y}{x}=\frac{3.5}{3}\approx1.17
eq1.5\). So not proportional.

Step3: Check Option B

For \( x = 1,y = 1.5\), \( \frac{y}{x}=1.5\); for \( x = 3,y = 4.5\), \( \frac{y}{x}=\frac{4.5}{3}=1.5\); for \( x = 6,y = 9.0\), \( \frac{y}{x}=\frac{9.0}{6}=1.5\); for \( x = 8,y = 12.0\), \( \frac{y}{x}=\frac{12.0}{8}=1.5\). And \( x = 0,y = 0 \). So this is proportional.

Step4: Check Option C

When \( x = 0,y = 0.5
eq0 \), so it's not a proportional relationship (proportional relationships must pass through the origin \((0,0)\)).

Step5: Check Option D

For \( x = 1,y = 0.5\), \( \frac{y}{x}=0.5\); for \( x = 3,y = 1.5\), \( \frac{y}{x}=\frac{1.5}{3}=0.5\); for \( x = 6,y = 3\), \( \frac{y}{x}=\frac{3}{6}=0.5\); for \( x = 8,y = 5\), \( \frac{y}{x}=\frac{5}{8}=0.625
eq0.5\). So not proportional.

Step1: Recall the formula for proportional relationship

The equation of a proportional relationship is \( y=kx \), where \( k=\frac{y}{x} \). We are given the point \((8,28)\) on the line.

Step2: Calculate the constant of proportionality \( k \)

Using the point \((x = 8,y = 28)\) in \( y=kx \), we have \( k=\frac{y}{x}=\frac{28}{8}=\frac{7}{2}=3.5 \).

Step3: Write the equation

Substitute \( k = 3.5\) into \( y = kx \), so the equation is \( y = 3.5x \) or \( y=\frac{7}{2}x \).

Step1: Recall the formula for proportional relationship

For a proportional relationship \( y = kx \), \( k=\frac{y}{x} \).

Step2: Calculate \( k \) using the first pair \((x = 6,y = 45)\)

\( k=\frac{45}{6}=7.5 \). Let's check with other pairs: for \( x = 10,y = 75\), \( \frac{75}{10}=7.5 \); for \( x = 12,y = 90\), \( \frac{90}{12}=7.5 \).

Step3: Write the equation

Since \( k = 7.5=\frac{15}{2}\), the equation is \( y = 7.5x \) (or \( y=\frac{15}{2}x \)).

Answer:

B

Question 10