Identify the transformations to the graph of a quadratic function
Change a quadratic function from general form to vertex form
Identify the axis and vertex of a parabola
2. Concepts and Objectives
Identify the transformations to the graph of a quadratic
function
Change a quadratic function from general form to vertex
form by completing the square
Identify the axis of symmetry and vertex of a parabola
using the vertex formula
3. Quadratic Functions
A function f is a quadratic function if
where a, b, and c are real numbers, and a 0.
The graph of a quadratic function is a parabola whose
shape and position are determined by a, b, and c.
2
f x ax bx c
4. Vertex Form
The graph of gx = ax2 is a parabola with vertex at the
origin that opens up if a is positive and down if a is
negative. The magnitude (or absolute value) of a
determines the width of the parabola.
The vertex form of a quadratic function is written
The graph of this function is the same as that of gx
translated h units horizontally and k units vertically.
This means that the vertex of F is at h, k and the axis of
symmetry is x = h.
2
F x a x h k
5. Vertex Form (cont.)
Example: Graph the function and give its domain and
range.
21
4 3
2
F x x
6. Vertex Form (cont.)
Example: Graph the function and give its domain and
range.
Compare to : h = 4 and k = 3 (Notice
the signs!)
Vertex: 4, 3, axis of symmetry x = 4
We can graph this function by graphing the base
function and then shifting it.
21
4 3
2
F x x
2
F x a x h k
7. Vertex Form (cont.)
Example, cont.:
Let’s consider the graph of
Vertex is at 0, 0
Passes through 2, ‒2 and
4, ‒8.
(I picked 2 and 4 because of
the half.)
21
2
g x x
8. Vertex Form (cont.)
Example, cont.:
To graph F, we just shift everything over 4 units to the
right and 3 units up.
Domain: ‒,
Range: ‒, 3] or y 3
9. Completing the Square
If we are given a function that is not in vertex form, we
can “complete the square” to transform it into vertex
form. We do this by taking advantage of the additive
identify property (a + 0 = a).
For example, the function is not a
binomial square. We can add 0 in the form of 52 – 52
(5 is half of 10), and group the parts that factor to a
binomial square:
22 2
10 305 5f x x x
2
10 30f x x x
2 2 2
10 5 5 30x x
2
5 5x
10. Completing the Square (cont.)
Example: What is the vertex of the function?
2
6 7f x x x
11. Completing the Square (cont.)
Example: What is the vertex of the function?
The vertex is at 3, ‒2.
2
6 7f x x x
22 2
3 736x x
2
3 9 7x
2
3 2x
6
3
2
12. Practice
Find the vertex by completing the square.
1. fx = x2 +8x + 5
2. gx = x2 – 5x + 8
3. hx = 3x2 + 12x – 5
13. Practice (cont.)
1. fx = x2 +8x + 5
The vertex is at ‒4, ‒11.
22 2
8 4 4 5f x x x
2
4 16 5x
2
4 11x
15. Practice (cont.)
3. hx = 3x2 + 12x – 5
The vertex is at ‒2, ‒17.
2
4 53h x x x
2 2 2
5243 3 2x x
2
3 2 12 5x
3 2 17x
Notice that I
factored the 3 out
of the first two
expressions.
16. Vertex Formula
You may have noticed that we are doing the same
process each time we complete the square to produce
the vertex form of the function. We can generalize this
to create a formula for the vertex of a parabola.
Starting with the general quadratic form, we can
manipulate it by completing the square and end up with
a version that gives us a formula for the coordinates of
the vertex.
17. Vertex Formula (cont.)
Deriving the vertex formula:
2
f x ax bx c
2 2
2 1 1
2 2
b b b
a x x c a
a a a
2 2
2 4
b b
a x c
a a
18. Vertex Formula (cont.)
Comparing this to the vertex form of
shows that
and
We can simplify this further by substituting h for x:
So for any quadratic function,
2
f x a x h k
2
b
h
a
2
4
b
k c
a
2
f h a h h k k
2
,f x ax bx c
, , and is the axis of symmetry.
2
b
h k f h x h
a
19. Vertex Formula (cont.)
Example: Find the axis and vertex of the parabola
having equation .
The advantage of the vertex formula here is that we
don’t have as many fractions to worry about:
2
2 4 5f x x x
4
2 2 2
b
h
a
4
1
4
axis: 1x
2
1 2 1 4 1 5f
2 4 5 3 vertex: 1,3
