Teach: Direct and Inverse Proportions ③ Graphs of Direct Proportion

Direct and Inverse Proportions ③ Graphs of Direct Proportion

When the equation of a direct proportion, y = ax, is represented as a graph, it becomes a straight line passing through the origin. The value of the constant of proportionality a determines the slope and direction of the line.

Graphs of Direct Proportion

The graph of y = ax is a straight line passing through the origin (0, 0). It changes according to the value of a.
　・a ＞ 0 → An upward-sloping line. As x increases, y also increases. It passes through the first and third quadrants.
　・a ＜ 0 → A downward-sloping line. As x increases, y decreases. It passes through the second and fourth quadrants.
　・The larger |a| is, the steeper the line is.

How to Draw the Graph

① Confirm that it passes through the origin.
② Calculate one point through which it passes (for example, y when x=1).
③ Connect the origin and the point found with a straight line.

For example, for the graph of y = 2x: when x = 1, y = 2, so draw the straight line passing through (0, 0) and (1, 2).

Example ①
State the quadrants through which each of the following direct-proportion graphs passes.
(1) y = 3x　(2) y = −2x　(3) y = (1/2)x
Solution:
(1) Constant of proportionality 3＞0 → first and third quadrants
(2) Constant of proportionality −2＜0 → second and fourth quadrants
(3) Constant of proportionality 1/2＞0 → first and third quadrants

Finding the Equation from a Graph

To find the equation from a graph of a direct proportion, read one point (x, y) other than the origin through which the graph passes, substitute it into y = ax, and find a.

Example ②
A graph of a direct proportion passes through the point (2, −6). Find its equation.
Solution:
Substitute x=2 and y=−6 into y = ax:
　−6 = 2a → a = −3
　Therefore, y = −3x
Key Points
  • The graph of y = ax is a straight line passing through the origin.
  • When a＞0, it slopes upward; when a＜0, it slopes downward.
  • To draw it, draw the line through the origin and (1, a) (if a is a fraction, another point may be used).
  • To find the equation from a graph, read one point on it and use a = y/x.

Practice Problems

  1. The graph of y = (1/3)x passes through the points (3, ?) and (−6, ?). Find the numbers that belong in the question marks.
  2. A graph of a direct proportion passes through the point (−4, 8). Find its equation.
  3. Compare the graphs of y = 2x and y = −x. Which line is steeper (has the greater slope)?

Answers and Explanations

  1. Answer: (3, 1) and (−6, −2)
    Explanation: In y = (1/3)x, x = 3 → y = 1, and x = −6 → y = −2.
  2. Answer: y = −2x
    Explanation: 8 = −4a → a = −2.
  3. Answer: y = 2x is steeper
    Explanation: The absolute values of the constants of proportionality are |2|=2 and |−1|=1. The greater value gives the steeper line, so y=2x is steeper.

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