Warm Up!

Complete the square.

3.1 Quadratic Functions and Models

2xxf

2

3

1xxf

23xxf

2

3

1xxf

2xxf 23xxf

If a > 0 the parabola opens up and the larger the a value the “narrower” the graph and the smaller the a value the “wider” the graph.

If a < 0 the parabola opens down and the larger the a the “narrower” the graph and the smaller the a the “wider” the graph.

2axxf

khxaxf 2

vertical shift, moves graph vertically by k

horizontal shift, moves graph horizontally by h

Determines whether the parabola opens up or down and how “wide” it is

162 xxxfWe need to algebraically manipulate this to look like the form above. We’ll do this by completing the square.

____1___62 xxxf

Add a number here to make a perfect square

Subtract it here to keep things equal (can’t add a number without compensating for it and we don’t want to add it to the other side because of function notation)

9 9

This will factor into (x-3)(x-3) so we can express it as something squared and combine the -1 and -9 on the end.

103 2 xxf

The graph of this function is a parabola

Let’s look at a quadratic function and see if we can graph it.

103 2 xxf

right 3

down 10

We started with and completed the square to get it in the format to be able to graph using transformations.

162 xxxf

We can take the standard form of the quadratic equation and do this to find a formula for the vertex.

What we find from doing this is on the next slide.

cbxaxxf 2

The x value of the vertex of the parabola can be found by computing

a

b

2

The y value of the vertex of the parabola can be found by substituting the x value of the vertex in the function and finding the function value.

162 xxxfLet’s try this on the one we did before:

a

bx

2 vertex of value

1

(1)

(-6) 3

1013633 vertex of value 2 fy

The vertex is then at (3, -10)

(3, -10)

Let’s plot the vertex: Since the a value is positive, we know the parabola opens up.

The parabola will be symmetric about a vertical line through the vertex called the axis or line of symmetry.

162 xxxf

Let’s find the y intercept by plugging 0 in for x.

10602 xf

So y intercept is (0, -1)

The graph is symmetric with respect to the line x = 3 so we can find a reflective point on the other side of the axis of symmetry.

(0, -1) (6, -1)

We can now see enough to graph the parabola

(3, -10)

Let’s look at another way to graph the parabola starting with the vertex:

We could find the x intercepts of the graph by putting f(x) (which is the y value) = 0

162 xxxfThis won’t factor so we’ll have to use the quadratic formula.

160 2 xx

So x intercepts are (6.2, 0) and (- 0.2, 0)

a

acbbx

2

42

12

11466 2

2.0 and2.62

406

A mathematical model may lead to a quadratic function. Often, we are interested in where the function is at its minimum or its maximum. If the function is quadratic the graph will be a parabola so the minimum (if it opens up) will be at the vertex or the maximum (if it opens down) will be at the vertex.

We can find the x value of the vertex by computing a

b

2

We could then sub this value into the function to find its minimum or maximum value.

xpR

4000,2002

1 xxp

DEMAND EQUATION

The price p and the quantity x sold of a certain product obey the demand equation:

This is the real world domain. The equation doesn’t make sense if the quantity sold is

negative (x < 0) and it doesn't make sense if the price is

negative (if x > 400)

Express the revenue R as a function of x.

Revenue is the amount you bring in, so it would be how much you charge (the price p) times how many you sold (the quantity x)

xxxxR 2002

1200

2

1 2

xxR 2002

1 2 This is a quadratic equation and since a is negative, its graph is a parabola that opens down. It will have a maximum value then at the y value of the vertex.

What is the revenue if 100 units are sold?

1002001002

1 2 R 000,15$

What quantity x maximizes revenue?

Since the revenue function is maximum at the vertex, we'll want to find the x value of the vertex to answer this.

200

21

2

200

2

a

bx

What is the maximum revenue?

This would be the y value of the vertex

2002002002

1200 2 f

000,20$

4000,2002

1 xxp

DEMAND EQUATION

The price p and the quantity x sold of a certain product obey the demand equation:

What price should the company charge to receive maximum revenue?

xxR 2002

1 2

Since we just found that the quantity to achieve maximum revenue was 200, we can substitute this in the price equation to answer this question.

100$2002002

1p

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au

Homework:pg. 189# 13, 17, 21, 25, 41, 43, 45, 59, 63 69

13, 17, 21 Get in vertex form (complete the square) then determine the transformations (shifting, compression or stretch and/or reflection)