Math 20 Pre-CalculusP20.7

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.

C. Quadratic Functions

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 FormP20.7 Demonstrate understanding of quadratic

functions of the form   y=ax²+bx+c and of their graphs, including:

vertexdomain and rangedirection of openingaxis of symmetryx- and y-intercepts.

1. Vertex FormInvestigate 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) = VertexOpens up +a Opens down –aLarger a = narrower parabola Smaller a = wider parabola

Example 1

Example 2

Example 3

Example 4

Key Ideas p.156

PracticeEx. 3.1 (p.157) #1-14

#4-18

2. Standard FormP20.7 Demonstrate understanding of quadratic

functions of the form   y=ax²+bx+c and of their graphs, including:

vertexdomain and rangedirection of openingaxis of symmetryx- 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 rightc 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

PracticeEx. 3.2 (p.174) #1-9, 11-17 odds

#5-25 odds

3. Completing the SquareP20.7 Demonstrate understanding of quadratic

functions of the form   y=ax²+bx+c and of their graphs, including:

vertexdomain and rangedirection of openingaxis of symmetryx- and y-intercepts.

3. Completing the SquareYou 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

PracticeEx. 3.3 (p.192) #1-9, 10-18 evens

#1-9 odds in each, 10-28 evens