Unit 8
8 marks · EasyA point P moves so that its coordinates (x, y) satisfy the quadratic equation .
Draw the graph of for .
Write down the coordinates of the points where the graph crosses the x-axis.
Let A be the point where on your graph. Write the vector , from the origin O(0,0) to A, as a column vector.
On the same grid, draw the vector , starting from the origin O(0,0).
You may estimate the y-value using the graph if needed.
Write down as a column vector.
What is the magnitude of ? Give your answer to 2 decimal places.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) — · must show Curve shape correct, Key points plotted: (–2,5), (0,–3), (1,–4), (2,–3), (5,12)
(b) (–1, 0) and (3, 0)
(c)
(d) — · must show Arrow from (0,0) to (2,–8)
(e)
(f) 4.12
Common mistakes: Mistaking the y-value at a point on the graph; Incorrect doubling of the vector; Wrong magnitude calculation; Mixing up x and y
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · Correct curve shape for for drawn.
B2 (a) · 1 mark · At least 3 correct points labelled or clearly plotted, including turning point and x-intercepts.
A1 (b) · 1 mark · Correctly identifies both coordinates where the curve crosses the x-axis: (–1, 0) and (3, 0).
M1 (c) · 1 mark · Writes as a column vector with x = 1, y = –4, or reads the y-value from the graph.
B3 (d) · 1 mark · Draws vector as (2,–8) from the origin on the grid.
B4 (d) · 1 mark · Correct length and direction for shown.
A2 (e) · 1 mark · Correct column vector stated.
A3 (f) · 1 mark · Correctly calculates the magnitude (), rounded to 2 decimal places.
Worked solution
a) Plot points for using :
| x | y |
|---|---|
| -2 | 5 |
| -1 | 0 |
| 0 | -3 |
| 1 | -4 |
| 2 | -3 |
| 3 | 0 |
| 4 | 5 |
| 5 | 12 |
Draw a smooth, U-shaped curve through all points from x = –2 to x = 5.
b) The curve crosses the x-axis where : Set . Factorise: so or . Thus, the crossing points are (–1, 0) and (3, 0).
c) At , . .
d) . Draw an arrow from (0,0) to (2,–8) on the same axes.
e) .
f) Magnitude of : (2 dp).