Factor and SolveQuadratic Equations

Ms. Nong

What is in this unit?

Graphing the Quadratic Equation Identify the vertex and intercept(s) for a parabola

Solve by taking SquareRoot & Squaring Solve by using the Quadratic FormulaSolve by Completing the SquareFactor & Solve Trinomials (split the middle)Factor & Solve DOTS: difference of two squareFactor GCF (greatest common factors)Factor by Grouping

The ROOTS (or solutions) of a polynomial are its x-intercepts

Recall: The x-intercepts occur where y = 0.

Roots

Roots ~ X-Intercepts ~ Zeros means the same

The number of real solutions is at most two.

Solving a Quadratic

No solutions

6

4

2

-2

5

f x = x2-2 x +56

4

2

-2

5

2

-2

-4

-5 5

One solution

X = 3

Two solutions

X= -2 or X = 2

The x-intercepts (when y = 0) of a quadratic function

are the solutions to the related quadratic equation.

Vertex (h,k)

Maximum point if the parabola is up-side-down

Minimum point is when the Parabola is UP

a>0 a<0

All parts labeled

Can you answer these questions?

How many Roots?

Where is the Vertex?(Maximum or minimum)

What is the Y-Intercepts?

What is in this unit?

Graph the quadratic equations (QE) Solve by taking SquareRoot & Squaring Solve by using the Quadratic FormulaSolve by Completing the SquareFactor & Solve Trinomials (split the middle)Factor & Solve DOTS: difference of two squareFactor GCF (greatest common factors)Factor by Grouping

Find the Axis of symmetry for y = 3x2 – 18x + 7

Finding the Axis of SymmetryWhen a quadratic function is in standard form

the equation of the Axis of symmetry is

y = ax2 + bx + c,

2ba

x This is best read as …

‘the opposite of b divided by the quantity of 2 times a.’

182 3

x 186

3The Axis of symmetry is x = 3.

a = 3 b = -18

Finding the VertexThe Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex.

STEP 1: Find the Axis of symmetry

Vertex

Find the vertex of y = -2x2 + 8x - 3

2ba

x a = -2 b = 8

x 82( 2)

8 4

2

X-coordinate

The x-coordinate of the vertex is 2

Finding the Vertex

STEP 1: Find the Axis of symmetry

STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.

8 8 22 2( 2) 4ba

x

The vertex is (2 , 5)

Find the vertex of y = -2x2 + 8x - 3

y 2 2 2 8 2 3 2 4 16 3

8 16 3 5

5

–1

( ) ( )22 3 4 3 1 5y= - - =

STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points

with a smooth curve.

y

x

( ) ( )22 2 4 2 1 1y= - - =-

3

2

yx

Graphing a Quadratic Function

Graph : y 2x 2 4x 1

y

x

Y-intercept of a Quadratic Function

y 2x 2 4x 1 Y-axis

The y-intercept of a

Quadratic function can

Be found when x = 0.

y 2x 2 4x 1

2 0 2 4(0) 10 0 1 1

The constant term is always the y- intercept

Example: Graph y= -.5(x+3)2+4

a is negative (a = -.5), so parabola opens down.Vertex is (h,k) or (-3,4)Axis of symmetry is the vertical line x = -3Table of values x y

-1 2 -2 3.5

-3 4 -4 3.5 -5 2

Vertex (-3,4)(-4,3.5)(-5,2)

(-2,3.5)(-1,2)

x=-3

Your assignment: