Slides for 2/9 & 2/10

Precalculus

Warm-Up Set 9 Problem 2

Using your calculator, find a line of regression for the following data set, and find the correlation coefficient r :

x y

0 6

2 9

4 16

6 18

8 20

10 25

12 31

14 34

Objectives

Today, we will: Graph a quadratic function by hand using

transformations and using technology. Express a quadratic function in the appropriate form:

Standard form Intercept form Vertex form

Find the intercepts (zeroes) of a quadratic function, and solve quadratic equations.

Graphing Quadratics

Use your calculator to graph the five example quadratics given here.

How are they alike?

How are they different?

234

1

234

1

23

3

2

2

2

2

2

xxp

xxh

xxg

xxf

xy

Standard Form for a Quadratic

cbxaxxf 2

In the standard form of a quadratic, the shape of the parabola is controlled by the parameters a, b, and c.

The y-intercept of the parabola is (0, c). If a is positive, the parabola opens upwards; if a is

negative, the parabola opens downwards. The line of symmetry is x = -b/(2a). A quadratic must be in standard form to use the

quadratic formula to find the x-intercepts.

Intercept Form for a Quadratic qxpxaxf The parameter a in the intercept form does the same thing it

did in the standard form. The x-intercepts of the parabola are (p, 0) and (q, 0) (watch

your signs!). The y-intercept of the parabola is (0, a·p·q). The axis of symmetry is x = (p + q)/2. We factor a quadratic to put it in intercept form – but some

quadratics cannot be factored! A quadratic function with no x-intercepts doesn’t have an intercept form.

Vertex Form for a Quadratic

khxaxf 2)( Again, a serves the same purpose in the vertex form

as in the two previous forms. The line of symmetry is x = h. The vertex of the parabola is (h, k). The y-intercept of the parabola is (0, a·h² + k). We can find the vertex form from the standard form

by completing the square.

Finding the Useful Parts When you graph a quadratic function, you must

always list the following important parts: The y-intercept The x-intercepts (if it has them; some parabolas do not) The line of symmetry The vertex

All three forms can give you the y-intercept and the line of symmetry.

To find the x-intercepts, either change to intercept form, or change to standard form and use the quadratic formula.

To find the vertex, either change to vertex form, or plug the x-value of the line of symmetry back into the quadratic function.

The Quadratic Formula

a

acbbx

2

42

Just in case you forgot, here’s the quadratic formula.

It finds the x-values of the x-intercepts of the parabola.

Since it requires a, b, and c, you have to be in the standard form to use the quadratic formula.