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iGCSE (2021 Edition)

4.06 Discriminant and parabolas

Worksheet
Quadratic equations and parabolas
1

For the following graphs of functions of the form y = a x^{2} + b x + c :

i

State whether the vertex of the parabola is a maximum or minimum point.

ii

State whether the value of a is negative or positive.

iii

State the number of solutions to the equation a x^{2} + b x + c = 0.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
2

Below is the result after using the quadratic formula to solve an equation:

x = \dfrac{- \left( - 10 \right) \pm \sqrt{ - 1 }}{8}

What can be concluded about the solutions of the equation?

3

By inspection, determine the number of real solutions for each of the following quadratic equations:

a
x^{2} = 9
b
\left(x - 4\right)^{2} = 0
c
\left(x - 6\right)^{2} = - 2
The discriminant and parabolas
4

When graphing a particular parabola, Katrina used the quadratic formula and found that b^{2} - 4 a c = - 5. How many x-intercepts does the parabola have?

5

When graphing a particular parabola, Tony used the quadratic formula and found that \\ b^{2} - 4 a c = 0. How many x-intercepts does the parabola have?

6

For each of the following graphs of a quadratic f \left( x \right) = a x^{2} + b x + c, with discriminant \\ \Delta = b^{2} - 4ac :

i

State whether a \gt 0 or a \lt 0.

ii

State whether \Delta \gt 0, \Delta \lt 0 or \Delta = 0.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
e
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
f
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
g
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
h
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
i
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
j
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
k
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
l
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
m
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
n
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
o
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
p
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
7

Consider the graph of the quadratic function:y = m - 9 x - 3 x^{2}

a

Find the possible values of m, if the graph has no x-intercepts.

b

State the largest possible integer value of m.

8

Determine the value(s) of k for which the graph of y = 4 x^{2} - 4 x + k - 15 just touches the \\ x-axis.

Points of intersection
9

Consider the parabola y = x^{2} - 8 x and the line y = - 7.

a

Form an equation to solve for the x-coordinate(s) of their point(s) of intersection.

b

Use the discriminant to determine how many points of intersection they have.

c

Find their point(s) of intersection.

10

Consider the parabola y = 6 x^{2} - 18 x + 51 and the line y = 18 x + 3.

a

Use the discriminant to determine how many points of intersection they have.

b

Find their point(s) of intersection.

11

Show that the line y = - 13 x - 10 is a tangent to the parabola y = \left(x - 3\right) \left(x - 2\right).

12

Consider the parabola y = - x^{2} - 3 x + 1 and the line y = - 3 x - 3.

a

Determine how many points of intersection they have.

b

Find their point(s) of intersection.

13

Consider the parabola y = x \left(x - 4\right) and the line y = - 3 x -5.

a

Form an equation to solve for the x-coordinate(s) of their point(s) of intersection.

b

Determine how many points of intersection they have.

14

Is the line y=5x-7 and tangent to the parabola y=4x^2-8x+10? Explain your answer.

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Outcomes

0606C2.1

Find the maximum or minimum value of the quadratic function f : x ↦ ax^2 + bx + c by any method.

0606C2.2

Use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain.

0606C2.3

Know the conditions for f(x) = 0 to have two real roots, two equal roots, no real roots. Know the related conditions for a given line to intersect a given curve, be a tangent to a given curve, not intersect a given curve.

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