Question

name london
uncommon schools | change history
7th grade math ia 3, l067 (proportional relationships)
date
team
proportionality in graphs
homework

1. on the lines below, explain how you know whether each graph represents a proportional relationship or not.

fig 1
fig 2
fig 3
fig 1: proportional? yes / no

fig 2: proportional? yes / no

fig 3: proportional? yes / no

does the following table show a proportional relationship? prove why or why not.
put (x) | 2 | 3 | 4 | 5
put (y) | 1 | 1.5 | 2 | 2.5

1. does the following table show a proportional relationship? prove why or why not.

input (x) | 2 | 3 | 4 | 5
output (y) | 4 | 5 | 6 | 7
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Explanation:

Step1: Analyze Fig 1

A proportional relationship graph is a straight line passing through the origin \((0,0)\). Fig 1 is a straight line starting at the origin, so it represents a proportional relationship. So Fig 1: YES.

Step2: Analyze Fig 2

A proportional relationship requires the line to pass through the origin and be straight with a constant rate. Fig 2 is a horizontal line (constant \(y\)-value) that does not pass through the origin (since it's parallel to the \(x\)-axis and above it), so it's not proportional. Fig 2: NO.

Step3: Analyze Fig 3

A proportional relationship graph must pass through the origin. Fig 3 is a straight line but does not pass through the origin (it starts above or below? Wait, looking at the graph, it's a line with negative slope but not passing through \((0,0)\) (since when \(x = 0\), \(y\) is positive, and it goes down). So it does not pass through the origin, so not proportional. Fig 3: NO.

Step4: Analyze First Table (put(x) and put(y))

For a proportional relationship, \(\frac{y}{x}\) should be constant. Calculate \(\frac{y}{x}\) for each pair:

• For \(x = 2\), \(y = 1\): \(\frac{1}{2}=0.5\)
• For \(x = 3\), \(y = 1.5\): \(\frac{1.5}{3}=0.5\)
• For \(x = 4\), \(y = 2\): \(\frac{2}{4}=0.5\)
• For \(x = 5\), \(y = 2.5\): \(\frac{2.5}{5}=0.5\)

Since \(\frac{y}{x}\) is constant (\(0.5\)), this table shows a proportional relationship.

Step5: Analyze Second Table (input(x) and output(y))

Calculate \(\frac{y}{x}\) for each pair:

• For \(x = 2\), \(y = 4\): \(\frac{4}{2}=2\)
• For \(x = 3\), \(y = 5\): \(\frac{5}{3}\approx1.666\)
• For \(x = 4\), \(y = 6\): \(\frac{6}{4}=1.5\)
• For \(x = 5\), \(y = 7\): \(\frac{7}{5}=1.4\)

Since \(\frac{y}{x}\) is not constant, this table does not show a proportional relationship.

Answer:

Fig 1: YES (because it's a straight line through the origin)
Fig 2: NO (horizontal line, not through origin)
Fig 3: NO (straight line, not through origin)
First Table: YES (constant \(\frac{y}{x}=0.5\))
Second Table: NO ( \(\frac{y}{x}\) is not constant)