CAIE A-Level Mathematics 1.3.5 Graph Intersections

Practise treating graph intersections as simultaneous solutions, eliminating a variable and using roots or discriminants to decide whether curves meet, touch or remain separate.

Exam points

equate graph formulas or eliminate one variable to form an equation for intersection coordinates
use the discriminant to prove that two graphs meet, touch or do not intersect
interpret real roots or counted crossings within the stated domain as equation solutions

[Maximum number: 2]

Figure for Question 1.3.5—Graph intersections question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=9\left(x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}\right)$ . The curve crosses the x-axis at the point A.

Find the x-coordinate of A.

$9\left(x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}\right)=0$ leading to $9 x^{-\frac{3}{2}}(x-4)=0$
M1
OE. Set y to zero and attempt to solve.
x=4 only
A1
From use of a correct method.

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1.3.5—Graph intersections question 1