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Standard Level

1.07 Vertex formula

Worksheet
Vertex Formula
1

Consider the parabola of the form y = a x^{2} + b x + c, where a \neq 0.

Complete the following statement:

The x-coordinate of the vertex of the parabola occurs at x = ⬚. The y-coordinate of the vertex is found by substituting x = ⬚ into the parabola's equation and evaluating the function at this value of x.

2

What does the line x = \dfrac{- b}{2 a} represent on the parabola defined by the equation \\y = a x^{2} + b x + c (a \neq 0)?

3

Consider the equation y = 6 x - x^{2}.

a

Find the x-intercepts of the quadratic function.

b

Find the coordinates of the turning point.

4

Consider the parabola y = x^{2} - 18 x + 10.

a

Is the graph of the function concave up or down?

b

State the value of x at which the minimum value of the function occurs.

c

State the minimum y-value of the function.

5

Consider the quadratic function y = x^{2} - 6 x + 5.

a

Is the graph of the function concave up or down?

b

Find the x-intercepts.

c

State the the y-intercept.

d

Determine the axis of symmetry.

e

Hence or otherwise state the coordinates of the vertex of the parabola.

6

Consider the quadratic function y = - x^{2} - 2 x + 8.

a

Is the graph of the function concave up or down?

b

Find the x-intercepts of the function.

c

Find the y-intercept.

d

Determine the axis of symmetry.

e

Hence or otherwise find the vertex of the curve.

7

Consider the quadratic function y = - 8 - 6 x - x^{2}.

a

Is the graph of the function concave up or down?

b

Find the x-intercepts of the function.

c

Find the y-intercept.

d

Determine the axis of symmetry.

e

Hence or otherwise find the vertex of the curve.

8

Consider the function y = x^{2} + 4 x + 3.

a

Determine the equation of the axis of symmetry.

b

Hence determine the minimum value of y.

c

Hence state the coordinates of the vertex of the curve.

d

Here is the graph of another quadratic function. State the coordinates of its vertex.

-5
5
x
-5
5
y
e

Determine whether the following relationship is true between the graph of \\y = x^{2} + 4 x + 3 and the graph provided.

i

They have the same x-intercepts.

ii

They share the same turning point.

iii

They have the same concavity.

9

Consider the function y = \left(14 - x\right) \left(x - 6\right).

a

State the zeros of the function.

b

Find the axis of symmetry.

c

Is the graph of the function concave up or concave down?

d

Determine the maximum y-value of the function.

10

Find the coordinates of the vertex of y = 3 x^{2} - 6 x - 9.

11

Find the maximum value of y for the quadratic function y = - x^{2} + 10 x - 25.

12

Consider the quadratic function f \left( x \right) = - 7.5 x^{2} - 1.3 x - 1.3.

a

Find the x-coordinate of the vertex correct to three decimal places.

b

Hence, find the maximum value obtained by the function, correct to two decimal places.

13

Find the x-coordinate of the vertex of the parabola represented by P \left( x \right) = p x^{2} - \dfrac{1}{2} p x - q.

14

Consider these two parabolas, labeled P_{1} and P_{2}.

\begin{aligned}P_{1}: y &= x^{2} + 4 x + 6\\P_{2}:y &= x^{2} - 4 x + 6\end{aligned}

a

Determine the coordinates of the vertex for each parabola.

b

How far apart are the vertices of the two parabolas?

15

Consider the curve y = x^{2} + 6 x + 4.

a

Determine the axis of symmetry.

b

Hence, determine the minimum value of y.

Applications
16

When an object is thrown into the air, its height above the ground is given by the equation h = 193 + 24 s - s^{2}where s is its horizontal distance from where it was thrown.

a

Find the value of s, at the point when it reaches its greatest height above the ground.

b

Find the maximum height reached by the object.

17

The height at time t of a ball thrown upwards is given by the equation h = 59 + 42 t - 7 t^{2}.

a

How long does it take the ball to reach its maximum height?

b

Find the height of the ball at its highest point.

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