Graph Proportional

Grade 8 · ratios · 100 practice problems · read aloud

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Graphing Proportional Relationships 📈

What is a Proportional Relationship?

A proportional relationship is when two quantities always have the same ratio. On a graph, this creates a straight line that always passes through the origin (0,0). This is useful for seeing how things change together, like speed and distance, or price and quantity.

How to Graph a Proportional Relationship

  1. Identify the constant of proportionality (k) from the equation y = kx.
  2. Plot the origin (0,0). All proportional lines start here!
  3. Use the constant (k) as your unit rate. From (0,0), move right 1 unit and up "k" units to plot your next point.
  4. Plot at least one more point using the same ratio (e.g., right 2, up 2k).
  5. Draw a straight line through the points, extending it in both directions.

Worked Examples

Example 1: Graph y = 3x

  1. Constant of proportionality, k, is 3.
  2. Plot (0, 0).
  3. From (0,0), move right 1, up 3 to plot (1, 3).
  4. Move right 1 more, up 3 more to plot (2, 6).
  5. Draw a straight line through the points.

Example 2: Graph the relationship where y is twice x.

  1. First, write the equation: y = 2x.
  2. Constant of proportionality, k, is 2.
  3. Plot (0, 0).
  4. From (0,0), move right 1, up 2 to plot (1, 2).
  5. Draw the straight line.

Common Mistakes to Avoid 🚫

  • Forgetting the origin: The line MUST go through (0,0). If it doesn't, it's not proportional!
  • Drawing a curved line: The graph must be a perfectly straight line.
  • Misreading the constant: In y = (1/2)x, from (0,0) you go right 1, up 1/2 (not right 1, up 2).

Tips & Tricks

  • The "Origin" Check: Before you start, ask: "If x is 0, is y also 0?" If yes, it's proportional!
  • Unit Rate is Key: The constant 'k' (like $3 per hour) tells you exactly how to move on the graph.
  • Table First: Make a quick table of (x, y) values to find points to plot.

How to Practice

  • Start by creating tables from equations like y = 4x before graphing.
  • Find real-world examples, like "If 1 book costs $5, graph the cost for multiple books."
  • Use online graphing tools to check your work after you draw it by hand.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 y = 6x = ?

Hint: In a proportional relationship y = kx, the constant k represents the slope. Think about what happens to y when x increases by 1.

Show the answer

Answer: 6

  1. The equation is y = 6x. This is in the form y = kx, where k is the constant of proportionality.
  2. The slope of a proportional relationship is equal to the constant k.
  3. Here, k = 6, so the slope is 6.
  4. As a unit rate, this means for every increase of 1 in x, y increases by 6.

The answer is 6.

2 (3² × 4) ÷ (6 - 2) = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication and division from left to right

Show the answer

Answer: 9

  1. Solve inside the parentheses: (6 - 2) = 4
  2. Calculate the exponent: 3² = 9
  3. Multiply: 9 × 4 = 36
  4. Divide: 36 ÷ 4 = 9

The answer is 9.

3 (-4, -12) and (2, 6) = ?

Hint: To find the slope between two points, consider how the vertical change relates to the horizontal change between them.

Show the answer

Answer: 3

  1. Recall the slope formula Slope m = (y2 - y1) / (x2 - x1)
  2. Assign coordinates Let (x1, y1) = (-4, -12) Let (x2, y2) = (2, 6)
  3. Substitute into the formula m = (6 - (-12)) / (2 - (-4)) m = (6 + 12) / (2 + 4) m = 18 / 6
  4. Simplify m = 3 So the slope is 3. Answer: 3

We are given two points: (-4, -12) and (2, 6). The problem likely asks for the slope of the line through these points.

4 (-3, -9) and (5, 15) = ?

Hint: Remember that slope represents the rate of change between two points on a line. Focus on the vertical change relative to the horizontal change.

Show the answer

Answer: 3

  1. Identify the coordinates: (x1, y1) = (-3, -9) and (x2, y2) = (5, 15)
  2. Apply the slope formula: slope = (y2 - y1) / (x2 - x1)
  3. Substitute the values: slope = (15 - (-9)) / (5 - (-3))
  4. Simplify the numerator: 15 - (-9) = 15 + 9 = 24
  5. Simplify the denominator: 5 - (-3) = 5 + 3 = 8
  6. Calculate the slope: 24 / 8 = 3

The answer is 3.

5 A line passes through points (2, 5) and (6, 17). Find its slope.

Hint: To find the slope of a line, use the formula that compares the vertical change to the horizontal change between two points.

Show the answer

Answer: 3

  1. Identify the coordinates of the two points. Point 1: (x1, y1) = (2, 5) Point 2: (x2, y2) = (6, 17)
  2. Substitute these values into the slope formula. slope = (17 - 5) / (6 - 2)
  3. Perform the subtraction in the numerator and the denominator. slope = (12) / (4)
  4. Simplify the fraction by dividing 12 by 4. slope = 3 Therefore, the slope of the line is 3.

To find the slope of a line that passes through two points, we use the slope formula: slope = (y2 - y1) / (x2 - x1)

6 A line passes through points (2, 5) and (6, 17). What is its slope?

Hint: To find slope between two points, use the formula that compares vertical change to horizontal change. Remember to subtract coordinates in the same order.

Show the answer

Answer: 3

  1. Identify the coordinates of the two points. Point 1: (2, 5) so x1 = 2 and y1 = 5 Point 2: (6, 17) so x2 = 6 and y2 = 17
  2. Substitute these values into the slope formula. slope = (17 - 5) / (6 - 2)
  3. Perform the subtractions inside the parentheses. slope = (12) / (4)
  4. Divide the numerator by the denominator. slope = 12 / 4 = 3 Therefore, the slope of the line is 3.

To find the slope of a line passing through two points, we use the slope formula: slope = (y2 - y1) / (x2 - x1)

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