Formulas, graphs & relations » Vertices of parabolas

Contents

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

Preface: how do you calculate y_vertex?

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

Symmetry

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

Example 1
It is known that a parabola has points (4, 13) and (12, 13). What is x_vertex?

Answer:
The axis of symmetry lies exactly between x = 4 and x = 12.
The x_vertex is on this axis of symmetry.
So x_vertex = 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 x_vertex = –3 + 72 = 2
Then y_vertex = 2(2 + 3)(2 – 7) = –50.
Coordinates are (2, –50).

x_vertex = –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:
x_vertex = –b2a

Example 1
Given is y = –4x² + 8x – 5.
Calculate the coordinates of the vertex of the corresponding graph.

Answer:
x_vertex = –82 × –4 = 1
Then y_vertex = –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 x_vertex:
x_vertex = –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 x_vertex = –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.
x_vertex = 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).

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