1.1 Direct Variation
In everyday life, many quantities change in relation to one another. For example, when the price per item is fixed, the total cost varies directly with the number of identical items bought. In direct variation, one quantity is proportional to another quantity or to a specified power of it; the corresponding ratio remains constant. This topic covers the definitions, algebraic and graphical representations, joint variations, and real-life problem-solving methods involving direct variations.
Learning Objectives
- Explain the meaning of direct variation.
- Determine the relation between two variables for a direct variation.
- Express direct variation involving powers or roots in variation notation and equation form.
- Determine the relation between three or more variables for a given joint variation.
- Solve problems involving direct variation and joint variation.
Characteristics of Direct Variation Graphs and Equations
Compare a direct variation graph with a non-direct linear relation, and recognise the forms \(y = kx\) and \(y = kx^n\).
Straight line passing through (0, 0), labelled y = kx, gradient = k.
Does not pass through origin ⇒ Not a direct variation.
\(y \propto x^n \Leftrightarrow \dfrac{y}{x^n} = k\) ⇔ Straight line of \(y\) against \(x^n\) through the origin.
Brief Explanation
Meaning of Direct Variation
Direct variation describes the relationship between two variables. For \(y \propto x\), the ratio \(\frac{y}{x}\) remains constant for nonzero \(x\). Multiplying \(x\) by a factor multiplies \(y\) by the same factor. This is written as “\(y\) varies directly as \(x\)”, or “\(x\) and \(y\) are directly proportional”.
Constant of Proportionality
A quantity \(y\) varies directly as \(x\) if and only if the ratio \(\frac{y}{x}\) is a constant. The constant of proportionality, k is a fixed or unchanged quantity value.
| Form | Variation relation | Equation form | Graph / Key Property |
|---|---|---|---|
| Direct variation | \(y \propto x\) | \(y = kx\) | Straight line of \(y\) against \(x\) through origin, gradient \(k\) |
| Power or root form | \(y \propto x^n\) | \(y = kx^n\) | Straight line of \(y\) against \(x^n\) through origin, gradient \(k\) |
| Joint variation | \(y \propto x^m z^n\) | \(y = kx^m z^n\) | The ratio \(\dfrac{y}{x^m z^n}\) is constant where \(x^m z^n \neq 0\) |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Given that \(y\) varies directly as the square root of \(x\), and \(y = 12\) when \(x = 16\). Express \(y\) in terms of \(x\).
The table shows \(x\): 2, 4, 6 and \(y\): 8, 32, 72. Determine whether \(y\) varies directly as \(x\) or as \(x^2\).
It is given that \(w\) varies jointly as \(p\) and the square of \(d\). If \(w = 54\) when \(p = 6\) and \(d = 3\), calculate \(w\) when \(p = 4\) and \(d = 5\).
Common Mistakes
- Assuming all straight-line graphs represent direct variation. The straight line must pass through the origin (0, 0). If \(c \neq 0\), it is not a direct variation.
- Forgetting powers or roots in equations. “Varies directly as the cube of \(x\)” must be written as \(y = kx^3\), not \(y = kx\).
- Inverting the constant of proportionality ratio. For \(y = kx\), calculate \(k\) as \(\frac{y}{x}\), not \(\frac{x}{y}\).
- Assuming a direct variation from only one pair of values. Compare multiple pairs to confirm that \(\frac{y}{x^n}\) is constant.
Misconceptions
Direct variation just means both variables increase together.
For \(y \propto x^n\), the ratio \(\frac{y}{x^n}\) must remain constant (for nonzero \(x\)).
If \(y \propto x^2\), the graph of \(y\) against \(x\) is a straight line.
The graph of \(y\) against \(x\) is a curve. Only the graph of \(y\) against \(x^2\) is a straight line through the origin.
Joint variation is an inverse variation.
Joint variation means a variable varies directly as the product of two or more variables.
Summary
- Direct variation between \(y\) and \(x\) means \(y \propto x\) and \(y = kx\), where \(k = \frac{y}{x}\) is the constant of proportionality.
- The graph of \(y\) against \(x\) for a direct variation is a straight line passing through the origin with gradient \(k\).
- For direct variation with powers or roots, \(y \propto x^n \Leftrightarrow y = kx^n\), where \(n = 1, 2, 3, \frac{1}{2}, \frac{1}{3}\). The graph of \(y\) against \(x^n\) is a straight line through the origin.
- Joint variation occurs when a variable varies directly as the product of two or more variables: \(y \propto x^m z^n \Leftrightarrow y = kx^m z^n\).
- To solve variation problems, find \(k\) using known values first; or equate ratios such as \(\dfrac{y_1}{x_1^{\,n}} = \dfrac{y_2}{x_2^{\,n}}\), or \(\dfrac{y_1}{x_1^{\,m} z_1^{\,n}} = \dfrac{y_2}{x_2^{\,m} z_2^{\,n}}\) for joint variation.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following statements is a necessary condition for variable \(y\) to vary directly as variable \(x\)?
2 It is given that \(y\) varies directly as the square of \(x\). If \(y = 32\) when \(x = 4\), what is the equation connecting \(y\) and \(x\)?
B. Fill in the Blanks
3 For the relation \(y = kx^n\), the value of \(k\) is known as the constant of .
4 When a variable varies directly as the product of two or more variables, this direct variation is known as a variation.
C. Drag and Drop: Match Variation Relation with Equation Form
Drag each relation to the correct equation. On a phone, tap the answer choice first, then tap the equation box.
