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Quadratic functions and parabolas

Dynamic parabola direction, vertex, axis, zeros, intercepts, graphs and coefficient effects.

Questions: 10Estimated time: 19 minutes

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Question 1out of 10

Give the vertex from the graph.

(3, -1)

(-1, -3)

(-3, 0)

(-2, -1)

(-3, 1)

(-3, -1)

Question 2out of 10

Give the vertex of y = 2(x − (-5))² + (-3).

(-5, -3)

(-5, -2)

(5, -3)

(-3, -5)

(-5, 3)

(-4, -3)

The vertex form of a parabola is y = a(x − m)² + n.

In this form, the vertex is V(m, n). Coefficient a does not change the vertex coordinates; it determines the opening direction and width.

Do not reverse the sign mentally: if the brackets contain x − (−m), the x-coordinate of the vertex is negative.

Question 3out of 10

Which way does the parabola y = 2x² open?

It has no branches

To the right

Upward

Downward

The sign of coefficient a determines the direction in which the parabola y = ax² opens:

when a > 0, it opens upwards;
when a < 0, it opens downwards.

The size of the coefficient changes the width of the parabola, but only its sign determines the opening direction.

Question 4out of 10

Where does y = 1x² + (1)x + (8) cross the y-axis?

(3, 1)

(4, 6)

(1, -4)

(3, -7)

(0, 8)

Question 5out of 10

Find the zeros of y = (x − (-8))(x − 8).

-7 and 9

-8 and -8

8 and 8

-8 and 8

Only -8

The zeros of a function are the x-values for which y = 0.

When the function is already factorised, use the zero-product rule:

(x − x₁)(x − x₂) = 0.

A product equals zero when at least one factor equals zero. Therefore solve x − x₁ = 0 and x − x₂ = 0 separately.

Question 6out of 10

Give the axis of symmetry of y = (x − (4))² + (-8).

y = 4

x = 4

y = -8

x = -8

For a parabola written as y = (x − m)² + n, the vertex is V(m, n).

The axis of symmetry is the vertical line passing through the vertex, so its equation is:

x = m.

The value n gives the height of the vertex but does not change the equation of the symmetry axis.

Question 7out of 10

Give the vertex of y = 4(x − (5))² + (1).

(1, 5)

(5, 2)

(-5, 1)

(5, 1)

(6, 1)

(5, -1)

The vertex form of a parabola is y = a(x − m)² + n.

In this form, the vertex is V(m, n). Coefficient a does not change the vertex coordinates; it determines the opening direction and width.

Do not reverse the sign mentally: if the brackets contain x − (−m), the x-coordinate of the vertex is negative.

Question 8out of 10

Where does y = -4x² + (4)x + (-5) cross the y-axis?

(-1, -6)

(0, -5)

(1, 11)

(-5, 2)

(2, -13)

Question 9out of 10

A ball’s height is modelled by a downward-opening parabola with coefficient −3. What does its vertex represent?

The graph has no maximum

The vertex represents a maximum height

The vertex represents the starting height only

Every point has the same height

Question 10out of 10

How does increasing |a| affect the width of y = ax²?

The value of a has no effect

A larger |a| makes the parabola narrower

A larger |a| always moves the vertex right

Coefficient a changes only the y-intercept

The graph of y = ax² is a parabola. The value |a| shows how strongly the parabola is horizontally compressed or stretched.

The larger |a| is, the narrower the parabola becomes.
The closer |a| is to zero, the wider the parabola becomes.

The sign of a determines its direction: when a > 0, the parabola opens upwards; when a < 0, it opens downwards.

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Sample questions

Which way does the parabola y = ___x² open?
Give the vertex of y = ___(x − (___))² + (___).
Give the axis of symmetry of y = (x − (___))² + (___).
Find the zeros of y = (x − (___))(x − ___).
Where does y = ___x² + (___)x + (___) cross the y-axis?