Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that:
(i) The graph of p(x) passes through the points (2, 3) and (6, 11).
(ii) The graph of q(x) passes through the point (4, −1).
(iii) The graph of q(x) is parallel to the graph of p(x).
Find the polynomials p(x) and q(x). Also, find the coordinates of the point where these lines meet the x-axis.
Step-by-step solution
Idea: Two points fix p. Parallel lines have the same slope, so q has slope a; one point then fixes d. On the x-axis, y = 0.
- p: 2a + b = 3 and 6a + b = 11. Subtracting, 4a = 8, so a = 2 and b = 3 − 4 = −1. p(x) = 2x − 1.1 mark
- q is parallel to p, so c = 2. Through (4, −1): 2 × 4 + d = −1 ⇒ d = −9. q(x) = 2x − 9.1 mark
- x-axis for p: 2x − 1 = 0 ⇒ x = 12, point (12, 0).1 mark
- x-axis for q: 2x − 9 = 0 ⇒ x = 92, point (92, 0). (Being parallel, the two lines never meet each other; each meets the x-axis at its own point.)1 mark
Check: p(2) = 3 ✓, p(6) = 11 ✓, q(4) = 8 − 9 = −1 ✓, and both slopes are 2 ✓.
Answer to write in the exam
2a + b = 3, 6a + b = 11 ⇒ 4a = 8 ⇒ a = 2, b = −1
p(x) = 2x − 1
q ∥ p ⇒ c = 2; q(4) = −1 ⇒ 8 + d = −1 ⇒ d = −9
q(x) = 2x − 9
2x − 1 = 0 ⇒ x = 12; 2x − 9 = 0 ⇒ x = 92
∴ p(x) = 2x − 1, q(x) = 2x − 9; x-axis points (12, 0) and (92, 0)
Common mistakes that cost marks
- Assuming parallel lines have the same y-intercept. They have the same slope; the intercepts differ.
- Working out the slope as 6 − 211 − 3 = 12 (upside down). Slope = rise ÷ run = 84 = 2.
- Giving the x-axis point as (0, −1) (that is where p cuts the y-axis).
How this can come in the exam
A line parallel to y = −3x + 7 passing through (1, 2) is
- y = −3x + 5
- y = 3x − 1
- y = −3x − 1
- y = −3x + 7
Show answer
(A) y = −3x + 5
Slope −3; 2 = −3 + d ⇒ d = 5.
The graph of p(x) passes through (1, 1) and (3, 7). Find p(x) and the linear polynomial q(x) whose graph is parallel to it and passes through the origin.
Show answer
2a = 6 ⇒ a = 3, b = −2: p(x) = 3x − 2 (2 marks). q(x) = 3x (1 mark).Try one yourself
Find the linear polynomial whose graph is parallel to that of 2x − 1 and passes through (−1, 4). Where does it cut the x-axis?
Show answer
2(−1) + d = 4 ⇒ d = 6: 2x + 6, cutting the x-axis at (−3, 0).
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