Pack A — Fluency

Mathematics

Year 11 · 11.4 Quadratic Functions

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
Solve $x^2 - 7x + 12 = 0$ by factorising.
2.
Solve $x^2 - 16 = 0$ .
3.
Solve $x^2 + 5x + 6 = 0$ using factorising.
4.
State the coordinates of the vertex of $y = (x - 3)^2 + -4$ and whether it is a min or max.
5.
State the $y$ -intercept of $y = x^2 - 5x + 7$ .
6.
Solve $2x^2 - 5x - 3 = 0$ by factorising.
7.
State the axis of symmetry of $y = x^2 - 6x + 8$ .
8.
Solve $x^2 - 4x - 5 = 0$ using the quadratic formula.
9.
Does $y = -2x^2 + 3x + 1$ open upwards or downwards?
10.
State the $x$ -intercepts of $y = (x - 2)(x + 5)$ .

SilverQuestions 11–20

11.
Express $x^2 + 8x + 10$ in the form $(x + p)^2 + q$ .
12.
Find the vertex of $y = x^2 - 6x + 11$ .
13.
Solve $2x^2 - x - 6 = 0$ by factorising.
14.
Expand $(x - 3)(x + 5)$ and write in $ax^2 + bx + c$ form.
15.
Write $y = 2(x - 3)^2 - 5$ in expanded form $y = ax^2 + bx + c$ .
16.
Solve $x^2 + 8x + 10 = 0$ exactly, leaving any irrational answer in surd form.
17.
A number plus its square is 30. Find the number.
18.
For $y = (x - 3)^2 - 4$ , state (a) the vertex, (b) the $x$ -intercepts.
19.
Express $y = 2x^2 - 8x + 5$ in vertex form.
20.
A rectangle has length $(x + 3)$ cm and width $x$ cm. Its area is 18 cm $^2$ . Find $x$ .

GoldQuestions 21–30

21.
Express $f(x) = x^2 + 8x + 10$ in vertex form, then state the range of $f$ .
22.
A quadratic $y = x^2 + bx + 7$ has its minimum value at $x = 3$ . Find $b$ .
23.
How many real roots does $2x^2 - 4x + 3 = 0$ have? Use the discriminant.
24.
The line $y = 2x + 1$ is tangent to the curve $y = x^2 + k$ . Find $k$ .
25.
Solve $x^2 - 5x + 6 \leq 0$ .
26.
A monic quadratic has roots $-2$ and $5$ . Write it in expanded form.
27.
A ball thrown follows $h(t) = -5t^2 + 20t + 1$ ( $h$ in m, $t$ in s). Find (a) the maximum height, (b) the time it occurs.
28.
For $x^2 + 5x - 6 = 0$ , find the sum and product of the roots without solving.
29.
A fountain jet has $h(x) = -(x - 4)^2 + 9$ ( $h$ height in m, $x$ horizontal distance in m). Find (a) max height, (b) range of $x$ above ground.
30.
For what range of $k$ does $x^2 + kx + 9 = 0$ have no real roots?

PlatinumQuestions 31–40

31.
A rectangle has perimeter 26 cm and area 40 cm $^2$ . Find its dimensions.
32.
For what values of $k$ does the line $y = x + k$ intersect $y = x^2$ at two distinct points?
33.
A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$ , $(1, 8)$ and $(3, 2)$ . Find $a$ , $b$ , $c$ .
34.
Express $y = x^2 - 6x + 11$ in vertex form. Hence describe the transformation from $y = x^2$ to $y = x^2 - 6x + 11$ .
35.
A parabola has vertex $(2, 3)$ and passes through $(5, 30)$ . Find its equation.
36.
Find the points of intersection of $y = x^2 + 2x - 3$ and $y = x + 1$ .
37.
Find the minimum value of $f(x) = 3x^2 - 12x + 7$ .
38.
Show that for all real $k$ , the equation $x^2 + 2kx + (k^2 + 1) = 0$ has no real solutions.
39.
A farmer has 40 m of fencing to enclose a rectangular pen against a wall (so one side is the wall). Find the dimensions giving the maximum area.
40.
A ball is thrown and its height (m) at times $t = 0, 1, 2$ is recorded as $1.5, 5, 7.5$ m. Assuming $h(t) = at^2 + bt + c$ , find $a$ , $b$ , $c$ and predict $h(3)$ .