Quadratics Functions-Vertex Form

Quadratic Functions

Vertex Form

Objectives

Students will be able to:

• Write quadratic functions in vertex form to represent relationships between

variables as shown in their graphs.

• Graph functions on coordinate axes using their key features.

• Interpret key features of the graph of a quadratic function.

Learning Targets Topic 2-1: Vertex Form of a Quadratic Function

A2.QFE.5: I can sketch the graph of a quadratic function given a verbal
description and show key features.

There are three commonly-used forms of quadratics:

Each quadratic form looks unique, allowing for different problems to be
more easily solved in one form than another. For this lesson we will be
looking at the vertex form of a quadratic. When in this form you can quickly
identify key features of the parabola and graph a sketch without using
technology. All quadratic functions are transformations of the parent
function y = x2. The vertex form of a quadratic function also shows how the
parent function can be transformed.

What Does the Vertex Form of a Quadratic Tell You? y = a(x-h)2 + k

The vertex is the point where the axis of symmetry intersects the parabola. It is also the lowest point of a parabola opening up or the highest point of a parabola opening down. As you may expect, the main benefit of vertex form is easily identifying

the vertex. The vertex of a parabola, or a quadratic equation, is written as (h,k) where the h is the x-coordinate

And the k is the y-coordinate. if a>0 (positive), the parabola opens upward and the vertex is a minimum. If a<0 (negative), the parabola opens downward, and the vertex is a maximum.

To find the axis of symmetry of a quadratic function in vertex form, simply look at the "h" value of the vertex, as the axis of symmetry will always be the vertical line equation "x = h" where (h, k) represents the vertex of the parabola.

The domain is all the x-values for which the function exists. Quadratic functions have a domain of all real numbers, written as (-∞,∞). This is a property of quadratic functions. You can plug any x-value into any quadratic function and you will find a
corresponding y-value.

The range is all the y-values for which the function exists. The range of a quadratic function is either from the minimum
y-value, k, to infinity, or from negative infinity to the maximum y-value, k. If the parabola opens upwards: The range is all
real numbers greater than or equal to the y-coordinate of the vertex , written as [k, ∞). If the parabola opens downwards:
The range is all real numbers less than or equal to the y-coordinate of the vertex, written as (-∞, k].

To find the y-intercept let x=0 and solve the equation for y

Identify a, h and k
a = 2 which is positive so the graph will face upward

h = 3 and k = 4 so the vertex is the point (3,4)

Because the parabola opens upward, the vertex is a minimum, the lowest point, the point where the parabola changes directions.

h = 3 so the equation for the axis of symmetry is x = 3. The axis of symmetry
is a vertical line that passes through the vertex. It acts as a mirror. All the
points on the left are the same distance from the line of symmetry as the
points on the right.

The domain is all real numbers; (-∞, ∞). The range is y≧4 also written as
[4, ∞)
Let x = 0 to find the y-intercept: y = 2(0-3)2 +4 = 2(-3)2 + 4 = 2(9) + 4 = 18 + 4
= 22. The y-intercept is (0, 22)

Graph the quadratic function f(x) = 2(x-3)2 + 4

1.

Plot the vertex (minimum)

2.

Draw the axis of symmetry

3.

Plot the y-intercept

4.

Plot another point the same distance from the axis of symmetry across from the y-intercept.

5.

Draw the U shaped parabola

A2.QFE.5: I can sketch the graph of a quadratic function given a verbal description and show key features.

(Narrow)

(Wider)

vertical

2

Q
U
A
D
R
A
T
I
C

Transforming Quadratic Functions Video Examples

a>1
0<a<1

Will the vertex be different?

A compression makes the graph __________
than the parent graph.

wider

What is the equation of the red graph?

What is the equation of the blue graph?

What is the equation of the green graph?

y = x2

y = 2x2 + 3

y = -½(x- 5)2

Describe the transformation from f(x)= x2:
1. y = -(x – 2)2 :

2. y = (-x)2 - 4

3. y = 2(x - 3)2

4. y = ½x2 + 1

Exit ticket