Formulas, graphs & relations » Vertices of parabolas

Contents

Preface: how do you calculate yvertex?
Symmetry
xvertex = b(2a)
How is xvertex = b(2a) derived?
Reasoning, finding (p, q) using the formula


Preface: how do you calculate yvertex?

In the theory below you will find an extensive explanation about finding the xvertex.
To calculate yvertex you first need to calculate xvertex.
Once you know xvertex, fill in the value of xvertex into the formula to calculate yvertex.
In function notation: yvertex = f (xvertex).

Symmetry

The quadratic function f (x) = a(x – α)(x – β) intersects the x-axis always in α and β.
So xvertex = α + β2 = lies precisely in between α and β.

In other words:
If you know two points that are exactly on the same height, then xvertex lies precisely in between.

Example 1

It is known that a parabola has points (4, 13) and (12, 13). What is xvertex?

Answer:
The axis of symmetry lies exactly between x = 4 and x = 12.
The xvertex is on this axis of symmetry.
So xvertex = 4 + 122 = 8.

Example 2

Calculate the coordinates of the vertex of the parabola that
corresponds to f (x) = 2(x + 3)(x – 7).

Answer:
From the function you can derive that the parabola will intersect the x-axis at x = –3 and x = 7. Solve the equation 2(x + 3)(x – 7) = 0, if you do not see this.
So xvertex = –3 + 72 = 2
Then yvertex = 2(2 + 3)(2 – 7) = –50.
Coordinates are (2, –50).

xvertex = b(2a)

The x-coordinate of the vertex of the graph that corresponds to function f (x) = ax2 + bx + c with a ≠ 0 can be calculated using this formula:
xvertex = b2a

Example 1

Given is y = –4x2 + 8x – 5.
Calculate the coordinates of the vertex of the corresponding graph.

Answer:
xvertex = –82 × –4 = 1
Then yvertex = –4 × 12 + 8 × 1 – 5 = –1.
Coordinates are (1, –1).

Example 2

Given is function f (x) = 2x2 + px + 3 of which the minimum is 1.
Calculate p algebraically.

Answer:
First make the formula for xvertex:
xvertex = p2 × –2 = –14p

If we substitute –14p for x in the formula than the outcome must be the minimum 1.
That gives us the following equation which can be solved:

2 × (–14p)2 + p × –14p + 3 = 1 
2 × 116p2 + –14p2 + 3 = 1 
18p2 – 14p2 + 3 = 1 
18p2 + 3 = 1 
18p2 = –2 
p2 = 16 
p = –4 of p = 4

How is xvertex = b(2a) derived?

Every quadratic equation can be written as ax2 + bx + c = 0.
However this equation does not always have solutions. The equation ax2 + bx + c = c always has solutions, as there always is an intersection with the vertical axis. At the first section of this page, you learned you can calculate the vertex using symmetry. So we have to take the average of the solutions of this equation.

ax2 + bx + c = c 
ax2 + bx = 0 
x(ax + b= 0
x = 0 of ax + b = 0
ax = –b
x = ba

Now we take the average of these to points that lie at the same height.
xvertex = 0 + ba2 = ba2 = b2a

Reasoning, finding (p, q) using the formula

The vertex of the graph corresponding to function f (x) = a(x – p)2 + q with a ≠ 0 is point (p , q). This is vecause when x = p you will have (x – p)2 = 0.

Example 1

Given is f (x) = 2(x – 3)2 – 4.
Give the coordinates of the vertex.

Answer:
p = 3 and q = –4 so the vertex is (3, –4).

Example 2

Given is f (x) = 7(3x + 9)2 + 8.
Give the coordinates of the vertex.

Answer:
Calculating p costs a little bit more work. You now have to solve 3x + 9 = 0.

 3x + 9 = 0
3x = –9
x = –3

p = –3 and q = 8 so the vertex is (–3, 8).


To top