Unit 5 Quadratics. Quadratic Functions Any function that can be written in the form

Unit 5 Quadratics

Quadratic Functions

• Any function that can be written in the form

Quadratic Functions

• Graph forms a parabola

• Label the parts of the parabola

or

To find the axis of symmetry

• When

Find the vertex and los

Vertex (h,k) form of a Quadratic

• Standard Form:

Parent Function

Transformations

• You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form

Sketch the graph

Sketch the graph

Sketch the graph

Sketch the graph

Sketch the graph

Identifying Important Parts on Calculator

• 2nd calc—then select max or min

Completing the Square

• Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve

• Steps:1. Move the constant2. Factor out the # in front of x2

3. Take ½ of middle term and square it4. Write in factored form for the perfect sq.

trinomial5. Add to both sides (multiply by # in front)6. Move constant back to get in (h,k) form

Examples

Examples

Example

Example

Example

Solving Quadratics

• You can solve by graphing, factoring, square root method, and quadratic formula

• Solutions, roots, or zeros

Solving by Graphing

1. Graph the parabola2. Look for where is

crosses the x-axis (where y=0)

3. May have 2 real, 1 real, or no real solutions

(Show on calculator)Review finding the

vertex

Solve the following by graphing

96.4

312.3

156.2

01032.1

2

2

2

2

xx

x

xx

xx

Solving Quadratics by Factoring

1. Factor the quadratic

2. Set each factor that contains a variable equal to zero and solve (zero product property)

More solving by factoring

253

168

1572

2

2

2

xx

xx

xx

You Try

Writing the Quadratic Eq.

• Write the quadratic with the given roots of ½ and -5

Write the quadratic with

• Roots of 2/3 and -2

When solving

• Graphing—not always best unless you have exact answers

• Factoring—not every polynomial can be factored

• Quadratic Formula—always works

• Square Root method—may have to complete the square first

Solving using Quadratic formula

• Must be in standard form

• Identify a, b, and c

a

acbbx

2

42

Examples

28122 xx

Examples

xx 221212

Examples

3242 2 xx

Examples

1342 xx

Discriminant

• Used to identify the “type” of solutions you will have (without having to solve)

numbertheoverradicalnoistherethatnote

acb

***

42

If the discriminant is…

• A perfect square---2 rational solutions

• A non perfect sq—2 irrational sol.

• Zero—1 rational sol.

• Negative—2 complex sol.

Identify the nature of the solution

04157.

0185.

2

2

xxex

xxex

Solving Quadratics using the Sq. Rt. method

• Useful when you have x2 = constant or a perfect sq. trinomial ex. (x-3)2=constant

1. Get the x2 by itself

2. Take the square rt. of both sides

3. Don’t forget + or – in your answer!!!

Examples

85.0164. 22 xexxex

Examples

22 216.653. xexxex

Examples

0253.0342. 22 xxexxxex

Quadratic Inequalities• Graphing quadratic inequalities in 2 variables:• Steps:• Graph the related equation• Test a point not on the graph of the parabola• Shade region that contains the point if it makes the

inequality true or shade the other region if it does not make the inequality true

• Ex. Ex. •

y x x 2 2 1 y x x 2 3 52

Graphing Quadratic Inequalities

542 xxy

Solving Quadratic Inequalities• Solving Quadratic Inequalities in one variable: May

be solved by graphing or algebraically.• To solve by graphing:• Steps:• Put the inequality in standard form• Find the zeros and sketch the graph of the related

equation• identify the x values for which the graph lies below the

x-axis if the inequality sign is < or • identify the x values for which the graph lies above the

x-axis if the inequality sign is > or

Solve by graphing

x x2 2 8 0 2 4 52x x

Solutions:_______________________ Solutions:_______________________

To solve algebraically:

• Steps:

• Solve the related equation

• Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set

• Test all regions of the number to determine other values to include in the solution set

Solve Algebraicallyx x2 3 18 x x2 11 30 0

4 12 102x x

Solving Quadratic Inequalities

2824 2 xx

Word Problems