standard form y = ax 2 + bx + c, where a ≠ 0 examples › y = 3x 2 › y = x 2 + 9 › y = x 2...

QUADRATICSWhat are they and why are they

important to me?

Uses of Quadratics

More Uses of Quadratics

Quadratic Functions Standard Form y = ax2 + bx + c, where a ≠ 0 Examples

› y = 3x2

› y = x2 + 9› y = x2 – x – 2 › y = - x2 + 2x - 4

Quadratic Parent Function

f(x) = x2

or

y = x2

Vocabulary Axis of symmetry

› The fold or line that divides the parabola into two matching halves.

Vertex› The highest or

lowest point of a parabola.

› Maximum or Minimum

Vocabulary (Continued)Dom

ainRange

The domain is all possible input (or x) values.› For our quadratics, the domain will always be

all real numbers.

The range is all output (or y) values.› For our quadratics, the range will always be

one of the following formats y > the y part of the vertex y ≥ the y part of the vertex y < the y part of the vertex y ≤ the y part of the vertex

Vocabulary

Vertex: (-2, 8)Axis of Symmetry:

x = -2MaximumDomain: All Real #’sRange: y ≤ 8

Graphing y = ax2 + c (no b)

a will determine whether there is a maximum or minimum value› If a > 0, then there is a minimum (parabola

opens up)› If a < 0, then there is a maximum

(parabola opens down) a also determines the “steepness” of

the quadratic function Vertex: the vertex will be (0, c)

Link to Examples of Graphing y = ax2 + c

Click here to go to examples and assignment

I created this power point using the 2010 version. If you are not using the 2010 version, you will need to go to my links and I have

included all links under the heading Quadratics.

y = ax2 + bx + c a still determines if maximum or

minimum a still determines “steepness” Axis of Symmetry: x = (equation of a vertical

line) Vertex: (, f()) f() just means to plug the x value of the

axis of symmetry into the quadratic to solve for y

Link to Examples of y = ax2 + bx + c

Click here for examples and assignments

I created this power point using the 2010 version. If you are not using the 2010 version, you will need to go to my links and

I have included all links under the heading Quadratics.

Solving Quadratic Equations

Standard Form of a quadratic equation: ax2 + bx + c = 0

Roots of the equation or zeros of the function› solutions of the quadratic equations› x-intercepts of the graph

Factoring to Solve Quadratic Equations Zero-Product

Property› For any real

numbers a and b, if ab = 0, then a = 0 or b = 0.

› Example: If (x+3)(x+2) = 0, then x+3 = 0 or x+2 = 0.

You can use the Zero-Product Property to solve quadratic equations of the form ax2+bx+C = 0 if that quadratic can be factored.

Remember to solve a quadratic equation means the same as finding the x-intercepts on the graph.

Solving Quadratic Equations

Click here for examples and assignments

I created this power point using the 2010 version. If you are not using the 2010 version, you will need to go to my links and I have included all links under the heading Quadratics.

Completing the Square

Turning x2 + bx into a perfect-square trinomial

Why do this?› Really the only reason to do this is to help

out when trying to find the vertex form of a quadratic function.

Completing the Square

Click here for examples and assignments

I created this power point using the 2010 version. If you are not using the 2010 version, you will need to go to my links and

I have included all links under the heading Quadratics.

Quadratic Formula

Solving Quadratic

Equations

Factoring and Zero-Product Property

Quadratic Formula

You should factor to solve a quadratic equation (find the x-intercepts) if the quadratic can be factored.

You are good at factoring.

You can use the quadratic formula to solve (find the x-intercepts) any quadratic equation.

You must memorize or at least know how to use the quadratic formula.

Solving Using the Quadratic Formula

Click for a link to the examples of using the quadratic formula

Practice Problems: Pages 571-572 #7-15, 29-34

I created this power point using the 2010 version. If you are not using the 2010 version, you will need to go to my links and I have

included all links under the heading Quadratics.

Pull it all together. Standard form of a quadratic function

› y = ax2 + bx + c Axis of Symmetry: x = (equation of a vertical line) Vertex: (, f())

Vertex form of a quadratic function› y = a(x – h)2 + k

Axis of Symmetry: x = h (equation of a vertical line) Vertex: (h, k)

Factored form of a quadratic function› y = a(x – r1)(x – r2)› x = r1 and x = r2 are possible x-intercepts

Key Ideas a in all forms will

› Determine the “steepness” of the parabola› Determine whether the parabola opens up

or down i.e. whether there is a maximum or minimum value

c is the y-intercept: › (0, c) is the point where the parabola

crosses the y-axis› not necessarily the vertex of the parabola.

Pull it ALL TogetherLinks