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Proportional
Graphs

Learn to identify proportional relationships from graphs and understand why they always pass through the origin.

Pre-Algebra

What Is a Proportional Relationship?

A proportional relationship between two quantities means that as one quantity changes, the other changes by a constant factor. The ratio between paired values always stays the same.

Visualizing Proportions on a Graph

When we plot a proportional relationship on a coordinate plane, something special happens: all the points fall on a straight line that passes through the origin \((0, 0)\).

0 1 2 3 4 5 2 4 6 8 10 \((0, 0)\) Apples Cost ($) \( y = 2x \)

Two Key Features of Proportional Graphs

★ Straight Line
All points lie on a perfectly straight line: no curves, no bends.
★ Through the Origin
The line must pass through \((0, 0)\). Zero input always gives zero output.

The Core Rule: \( y = kx \)

Every proportional relationship can be written as:

\( k \) is the constant
The constant of proportionality. It tells you how much \( y \) changes for every 1 unit of \( x \).
\( k \) is the slope
On the graph, \( k \) equals the slope of the line. Steeper line means larger \( k \).

Why Must It Pass Through the Origin?

1
Zero in means zero out. If you buy 0 apples, you pay $0. If you drive for 0 hours, you travel 0 miles.
2
The equation confirms it. Substituting \( x = 0 \) into \( y = kx \) always gives \( y = 0 \).
3
A starting fee breaks it. If a taxi charges $5 just to get in, the cost starts at $5 when distance is 0: that's \((0, 5)\), not \((0, 0)\).

Proportional vs. Not Proportional

Proportional \((0,0)\) x y ✓ \( y = kx \) Not Proportional \((0,3)\) x y ✕ \( y = -x + 3 \)

Linear vs. Proportional

Linear Proportional
Equation \( y = mx + b \) \( y = kx \)
Graph Shape Straight line Straight line
Passes Through Origin? Only if \( b = 0 \) Always
Constant Ratio \( y/x \)? Not always Always
Example \( y = 3x + 5 \) \( y = 3x \)

How to Identify a Proportional Graph

1
Check the origin. Does the line (or do the points) pass through \((0, 0)\)? If not, stop: it's not proportional.
2
Check for a straight line. Can you draw one straight line through every point? If the points curve, it's not proportional.
3
Verify the constant ratio. Pick any point \((x, y)\) on the line and compute \( y \div x \). It should be the same for every point.
4
State the relationship. Write \( y = kx \) with your constant \( k \). That's the equation of the proportional line.

Example 1: Distance vs. Time

A car travels at a constant speed. The graph below shows distance over time. Is this proportional?

0 1 2 3 4 60 120 180 240 Time (hours) Distance (mi) \((0,0)\)

Example 2: Gym Membership

A gym charges a $20 sign-up fee plus $10 per month. The graph plots total cost vs. months. Is this proportional?

0 1 2 3 4 10 20 30 40 50 60 \((0, 20)\) origin Months Cost ($)

Example 3: Is a Curve Proportional?

The area of a square with side length \( s \) is \( A = s^2 \). The graph passes through the origin. Is this proportional?

0 1 2 3 4 4 8 12 16 \((0,0)\) Side length (s) Area

Finding \( k \) from a Graph

Once you confirm a graph is proportional, you can find \( k \) by picking any point \((x, y)\) on the line (other than the origin) and computing:

0 2 4 6 6 12 18 \((2, 6)\) \( k = \frac{6}{2} = 3 \) x y

Comparing Different Constants of \( k \)

0 1 2 3 4 2 4 6 8 \( k = 1 \) \( k = 2 \) \( k = 4 \) Steeper = larger k x y
Small \( k \)
Shallow line: \( y \) grows slowly relative to \( x \).
Large \( k \)
Steep line: \( y \) grows quickly relative to \( x \).

Common Mistakes

✕ Ignoring the Origin
Seeing a straight line and calling it proportional without checking that it passes through \((0, 0)\).
✕ Curve Through Origin
A curve like \( y = x^2 \) goes through the origin but is not a straight line, so it's not proportional.
✕ Inconsistent Ratios
Checking only one point's ratio. You must verify that \( y/x \) is the same for every point on the line.

Quick Reference Summary

Feature Proportional Not Proportional
Graph Shape Straight line Curve or straight line
Passes Through \((0,0)\)? Always yes Not necessarily
Equation Form \( y = kx \) \( y = mx + b \) or nonlinear
Ratio \( y/x \) Always constant Changes
Real Example $3 per lb of fruit $5 base + $3 per lb

Question 1 of 5

A graph shows a straight line that passes through the points \((0, 0)\) and \((3, 12)\). Which equation represents this relationship?

Question 2 of 5

Which of these graphs shows a proportional relationship?

Question 3 of 5

A proportional graph passes through the point \((5, 35)\). What is the constant of proportionality \( k \)?

Question 4 of 5

A graph shows the equation \( y = 6x + 2 \). Why is this relationship not proportional?

Question 5 of 5

A table shows these pairs: \((1, 4)\), \((2, 8)\), \((3, 12)\), \((4, 16)\). If graphed, would this be proportional?