Substituting into f(x) · solving for a constant · reading rational-function graphs
Fixes these misses: M2 Q6 (solve for a constant), M2 Q18 (rational-function graph).
Core concepts
Evaluating and solving
f(a) means substitute a for x everywhere, then simplify.
If f(a) = value, set the expression equal to that value and solve for a step by step.
Rational functions & asymptotes
Vertical asymptote: where the denominator = 0 (the graph shoots to ±∞).
Horizontal asymptote: the value the graph approaches as x → ±∞ — often the ratio of leading coefficients, or y = 0 when the bottom grows faster.
Match a described graph to an equation by checking: where does it blow up? what does it level off to?
Reading a described curve
"Approaches the line x = −4" → vertical asymptote at x = −4 → denominator has (x + 4).
"Approaches y = 0 as x decreases" → horizontal asymptote y = 0.
Worked example — solve for a constant
Problem. f(x) = (x + 15)/5 and f(a) = 10. Find a.
Substitute: (a + 15)/5 = 10.
Multiply by 5: a + 15 = 50.
a = 35. (Mirrors the structure of M2 Q6.)
Watch out: When a function has a fraction, clear the denominator first by multiplying both sides — don’t try to eyeball it. And match asymptotes to the graph before checking specific points; the vertical asymptote alone often eliminates 2–3 choices.
Video lessons
Curated walkthroughs from trusted math channels — watch one or two before doing the worksheet.