1718 WH3X CH2 Lesson 5

• Wiskunde
• Middelbare school
• havo
• Leerjaar 3

Finding the vertex

(without spending a lot of time plotting the graph...)

Vertex

The vertex is the highest (downward opening parbola)
or lowest (upward opening parabola) point of a parabola.

The vertex lies on the axis of symmetry
Therefore, the x-coordinate of the vertex is the same as the x-value of the axis of symmetry.

Axis of symmetry

Since the axis of symmetry divides the parabola into two equally shaped halves, the x-value of the axis of symmetry can be found halfway between two points on the parabola with the same y-value.

(-3,0) and (-1,0) have the same
y-value. The vertex is then in the
middle of x = -3 and x = -1, so at x = -2.
(Notice that that is also
between x = -4 and x = 0)
Fill in -2:
The vertex is then at (-2, -1)

y=x​2​​+4x+3

y=(−2)​2​​+4⋅−2+3=−1

(1,0) and (3,0) have the same
y-value. The vertex is then in the
middle of x = 1 and x = 3, so at x = 2.
(Notice that that is also
between x = 0 and x = 4)
Fill in 2:
The vertex is then at (2, -0.5)

y=0.5x​2​​−2x+1.5

y=0.5⋅2​2​​−2⋅2+1.5=−0.5

(0,-3) and (3,-3) have the same
y-value. The vertex is then in the
middle of x = 0 and x = 3, so at x = 1.5
(Notice that there are no points of
intersection with the horizontal axis)
Fill in 1.5:
The vertex is then at (1.5, -0.75)

y=−x​2​​+3x−3

y=−1⋅1.5​2​​+3⋅1.5−3=−0.75

(-5,-2) and (0,-2) have the same
y-value. The vertex is then in the
middle of x = -5 and x = 0, so at x = -2.5
(Notice that the points of intersection
with the x-axis are not easy to use)
Fill in -2.5:
The vertex is then at (-2.5, 4.25)

y=−x​2​​−5x−2

y=−1⋅(−2.5)​2​​−5⋅−2.5−2=4.25

Calculating the vertex

1) Find two points on the parabola with the same y-value (points of intersection with the x-axis or with the horizontal line through the point of intersection with the y-axis).
2) Find the x-value of the axis of symmetry in the middle of those two points.
3) Fill in the x-value in the formula to calculate the y-value
4) Write down the coordinates of the vertex

Example

Calculate the coordinates of the vertex of the parabola with formula:
1) If x = 0, y = -8, so point of intersection y-axis = (0, -8)



                               or

 

y=3x​2​​+12x−8

3x​2​​+12x−8=−8

3x​2​​+12x=0

3x(x+4)=0

3x=0

x=0

x+4=0

x=−4

Example

1)                          or
2) So axis of symmetry is at
3) Fill in:
4) Coordinates vertex are: (-2, -20)

 

x=0

x=−4

x=−2

y=3⋅(−2)​2​​+12⋅−2−8=−20

y=3x​2​​+12x−8

Homework Tuesday

§2.2:
Make ALL key exercises: 10, 13, 14, 15
Make AT LEAST 1 from the basic: 11, 12 or advanced exercises: 16
Correct your answers