Factor and Solve Quadratic Equations
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Transcript of Factor and Solve Quadratic Equations
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: