C. Quadratic Functions
Math 20 Pre-Calculus
P20.7
Demonstrate understanding of quadratic functions
of the form y=ax²+bx+c and of their graphs,
including: vertex, domain and range, direction of,
opening, axis of symmetry, x- and y-intercepts.
Key Terms:
 Quadratic Functions occur in a wide variety of real world
situations. In this unit we will investigate functions and use
them in math modelling and problem solving.
1. Vertex Form
 P20.7
 Demonstrate understanding of quadratic functions of the





form y=ax²+bx+c and of their graphs, including:
vertex
domain and range
direction of opening
axis of symmetry
x- and y-intercepts.
1. Vertex Form
 Investigate
p. 143
 The graph of a Quadratic Function is a parabola
 When the graph opens up the vertex is the lowest point and
when it opens down the vertex is the highest point
 The y-coordinate of the vertex is called the min value or max
value depending of which way it opens.
 The parabola is symmetrical about a line called the axis of
symmetry. The line divides the graph into two equal halves,
left and right.
 So if you know the a of s and a point you can find another
point (unless the point is the vertex)
 The A of S intersects the vertex
 The x-coordinate of the vertex is the equation of the A of S.
 Quadratic Function in vertex form f(x) = a(x-p)2+q are
very easy to graph.
 a, p, and q tell you what you need.
 (p,q) = Vertex
 Opens up +a
 Opens down –a
 Larger a = narrower parabola
 Smaller a = wider parabola
Example 1
Example 2
Example 3
Example 4
Key Ideas
p.156
Practice
 Ex. 3.1 (p.157) #1-14
#4-18
2. Standard Form
 P20.7
 Demonstrate understanding of quadratic functions of the





form y=ax²+bx+c and of their graphs, including:
vertex
domain and range
direction of opening
axis of symmetry
x- and y-intercepts.
2. Standard Form
 Recall that the Standard form of a quadratic function is
f(x) = ax2+bx+c
or
y = ax2+bx+c
 Where a, b, c are real numbers and a ≠ 0
 a determines width of graph (smaller a = wider graph) and
opening (+a up and –a down)
 b shifts the graphs left and right
 c shifts the graph up and down
 We can expand f(x) = a(x-p)2+q to get f(x) = ax2+bx+c ,
which will allow us to see the relation between the variable
coefficients in each.
 So,
 b = -2ap
or
And
 c = ap2 + q
or q = c – ap2
 Recall that to determine the x-coordinate of the vertex, you
use x = p.
 So the x-coordinate of the vertex is
Example 1
Example 2
Example 3
Key Ideas
p.173
Practice
 Ex. 3.2 (p.174) #1-9, 11-17 odds
#5-25 odds
3. Completing the Square
 P20.7
 Demonstrate understanding of quadratic functions of the





form y=ax²+bx+c and of their graphs, including:
vertex
domain and range
direction of opening
axis of symmetry
x- and y-intercepts.
3. Completing the Square
 You can express a quadratic function in vertex form, f(x) =
a(x-p)2+q or standard form f(x) = ax2+bx+c
 We already know we can go from vertex to standard by just
expanding
 However to graph by hand it is much easier if the function is
in vertex form because we have the vertex, axis of symmetry
and max or min of the graph
 So to be able to turn a standard form function into vertex
form would be advantageous.
 This process is called Completing the Square
 What we want to be able to do is rewrite the trinomial as a
binomial squared. (x+5)(x+5) = (x+5)2
 Lets complete the square:
 If there is a coefficient in front of the x2 term we have to add
a couple steps.
Complete the Square:
Example 1
Example 2
Example 3
Example 4
Key Ideas
p.192
Practice
 Ex. 3.3 (p.192) #1-9, 10-18 evens
#1-9 odds in each, 10-28 evens