Quadratics
Quadratics questions to try without a calculator
1
Determine the nature of the roots of the function
f(x) = 3x² + 4x – 2
2 Marks
Show Answer
b² – 4ac =16
2 real and distinct roots
2
a Write the equation y = x² – 8x + 19 in the form y = (x – a)² + b
2 Marks
b Hence, sketch the graph of y = x² – 8x + 19 in the form.
On your graph show clearly the coordinates of the turning point and the point
of intersection with the y-axis.
3 Marks
Show Answer
a y = (x – 4)² + 3
b Graph of positive parabola with min TP at (4, 3) and y intercept at (0, 19)
3
The diagram shows the graph of y = kx²
Calculate the value of k
2 Marks
Show Answer
k = 5
4
Determine the nature of the roots of the function
f(x) = 2x² + 7x – 5
2 Marks
Show Answer
b² – 4ac = 89
2 real and distinct roots
5
The diagram shows the graph of y = kx²
Calculate the value of k
2 Marks
Show Answer
k =3
6
The diagram shows the graph of y = kx²
Calculate the value of k
2 Marks
Show Answer
k =- 3
7
Determine the nature of the roots of the function
3x(x – 2) = – 5
3 Marks
Show Answer
no real roots
8
Given that f(x) = x² – 3x
Determine the value of f(–6)
3 Marks
Show Answer
54
9
a Write the equation y = x² +10x + 27 in the form y = (x + p)² + q
2 Marks
b Hence, sketch the graph of y = x² +10x + 27 in the form.
On your graph show clearly the coordinates of the turning point and the point
of intersection with the y-axis.
3 Marks
Show Answer
a (x + 5)² + 2
b Positive parabola with TP (–5, 2) and y intercept (0, 27)
10
The diagram below shows the path of a small rocket which is fired into the air.
The height, h metres of the rocket after t seconds is given by
h(t) = 10t – t²
a After how many seconds does the rocket first reach 16m?
4 Marks
b Will the rocket reach a height of 24m.
Justify your answer
2Marks
Show Answer
a 2 seconds
b yes
11
The diagram below shows the graph of y = (x + a)² + b
The axis of symmetry is x = 5
a State the value of a
1 Marks
The point (9, 23) lies on the parabola.
b Determine the value of b
2 Marks
Show Answer
a a=–5
b b = 7
12
f(x) = x² + 2x
g(x) = 11x – 20
f(x) = g(x) at points A and B
a Show that x² – 9x + 20 = 0 at points A and B
2Marks
b Hence, determine algebraically the x values of A and B
2 Marks
Show Answer
b x = 4, 5
13
The logo for the Upside company is in the shape of a parabola.
The parabola has equation y = x² – 8x + 12
a Determine the coordinates of A and B
3 Marks
b Hence, calculate the height of the letter U
3 Marks
Show Answer
(2, 0) and (6, 0)
16 units
14
A science teacher fire a rocket from the roof of his school.
The diagram shows the path of the rocket
The height h m is given by
h(t) = 24 + 10t – t²
a How high is the roof of the school?
1 Mark
b Determine, algbraically the time taken for the rocket to hit the ground.
4 Marks
Show Answer
a 24m
b 12s
15
The diagram shows the graph of y = (x + a)² + b
State the values of a and b
2 Marks