A linear-law graph is a tool for learning about the original relation. The last marks come from converting back: the constants, the predicted values and a sentence saying what they mean.
This lesson closes the linear law chapter after finding constants from gradient and intercept.
How do I convert back to the original variables?
Write down what Y and X stood for, then reverse the conversion. If X = x², then x = √X. If Y = lg y, then y = 10^Y. The line is only a tidy route to the original equation, not the answer in itself.
Worked example: a machine losing value
A machine’s value V (in RM) after t years follows V = a·b^t. A graph of lg V against t is a straight line through (0, 4.0) and (5, 3.5).
The linear form is lg V = (lg b)t + lg a. So the intercept lg a = 4.0 gives a = 10 000.
The gradient is (3.5 − 4.0) ÷ 5 = −0.1, so lg b = −0.1 and b = 10^−0.1 = 0.794 (3 s.f.). The model is V = 10 000 × 0.794^t.
Meaning: a is the starting value, RM10 000. A b of 0.794 means the machine keeps about 79.4% of its value each year, losing about 20.6% yearly.
Predicting in both directions
Forward: find V when t = 10. Use the line: lg V = 4.0 − 0.1(10) = 3.0, so V = 10^3 = RM1 000.
Backward: when is V = RM5 000? Then lg 5 000 = 3.699, so 4.0 − 0.1t = 3.699, which gives t = 3.01 years.
A second model: average cost
The average cost C (in RM) of making x items is C = a/x + b. A graph of C against 1/x passes through (0.01, 7) and (0.05, 11).
Gradient: (11 − 7) ÷ (0.05 − 0.01) = 100, so a = 100. Substitute (0.01, 7): 7 = 100(0.01) + c, so c = 6 and b = 6.
Meaning: a = 100 is a fixed cost shared across all items, and b = 6 is the cost of making one more item. As x grows large, C approaches 6 and never goes below it.
The mistake that costs marks
A common slip is to read a value directly from the graph and report it as the original quantity. If the question gives x = 4 and the graph is y against x², the correct input on the graph is X = 16, not X = 4.
The second slip is leaving the answer as lg V. A reader who sees “3.0” has no way to know the machine’s value is RM1 000.
Check yourself
A graph of lg y against lg x is a straight line through (0, 0.70) and (1, 2.70). The relation is y = a·x^n. Find a and n, then find y when x = 3.
Answer
The linear form is lg y = n(lg x) + lg a. The intercept is 0.70, so lg a = 0.70 and a = 10^0.70 = 5.01.
The gradient is (2.70 − 0.70) ÷ 1 = 2, so n = 2. The relation is y = 5.01x².
When x = 3: y = 5.01 × 9 = 45.1. Check by the line: lg y = 0.70 + 2 lg 3 = 1.654, and 10^1.654 = 45.1.
What to study next
Test the whole chapter with the linear law practice set. The linear law axes explorer lets you see how a chosen Y and X change the graph. Use the mistake log to note which step went wrong.
If you want a teacher to rehearse context explanations with you, see online one-to-one Additional Mathematics tuition.