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Quadratics
By: Aditi Singh, Maegan Lloyd, Bella
4th period
Un-Factoring
Methods of Un-factoring:
1.FOIL
2.Box Method
Un-Factoring:FOIL
First
(x+a)(x+b)
Outer
x2+ax+bx+ab
Inner
x2+(a+b)x+(ab)
Last
Un-Factoring:Box Method
x
x
x2
bx
a
ax
ab
x2+ax+bx+ab
x2+(a+b)x+ab
b
Un-Factoring:Example
1.(x-9)(x+2)
(2x+4)(x-5)
2.
Factoring
x2+bx+c
b
Factor
Factor
Factor
+
Factor
Factor
Factor
c
Factor
*
Factor
➔What 2 numbers do you add to get b and multiply to get c?
Factoring:Example
1.x2-7x-18
Factoring:With Co-Efficient
ax2+bx+c=0
b
Factor
Factor
Factor
+
Factor
Factor
(ax2+(factor)x)
(+(factor)x+c)
Factor
ac
Factor
*
Factor
**Grouping method
**will make more sense
in example
➔What 2 numbers do you add to get b and multiply to get
a*c?
Factoring:Example
4x2-19x+12=0
**want to take out same
factor from both
Factoring: Worksheet Example
7x2 + 9x
Zeros
Zero:x-intercepts of graph; when y=0
(x+a)=0
(x+b)=0
**solve for x
Zeros:Example
1.(7x+4)(x-7)=0
2. 5x2-x-16=2
Zero: Worksheet Example
5r2
− 44r + 120 = −30 + 11r
Quadratic Formula
**x2+bx+c
➔Use Quadratic Formula to find x-intercepts when equation
is unfactorable
➔Simply plug in values into equation and solve
Quadratic Formula:Example
9x2+7x-56
Quadratic Formula: Worksheet Example
5x2 + 9x = −4
Discriminant
b2-4ac
If discriminant is:
Negative-no real solutions to equation
0-1 real solution to equation
Positive-2 real solutions to equation
**Solution:x-intercept
Discriminant: Example
1.Discriminant of: 9x2+7x-56
2.Number of solutions:7x2-32x-6
Discriminant: Worksheet Example
−9b2 = −8b + 8
AOS
Discriminant
Axis of Symmetry (AOS)
x=(-b)/(2a)
Maximum
Maximum- highest point on the graph
Minimum
Minimum- lowest point on the graph
Vertex
Vertex- the maximum or minimum of a graph
How to find vertex:
1.Find axis of symmetry (x value)
2.Plug in that value into quadratic equation (y value)
3.Coordinates (x, y)
Vertex:Example
Find vertex & state if max/min: x2+5x+7
Vertex: Worksheet Example
y = -2x2 + 4x + 1
Graphing Quadratics
1.Calculate and graph Axis of Symmetry (points can be
reflected over AOS)
2.Calculate and graph vertex
3.Calculate and graph zeros
4.Calculate and graph y-intercepts
5.Plug in other x-values into equation to complete the
shape of parabola
Graphing Quadratics:Example
x2-9x+8
Quadratic Inequalities
< or > use dotted line
< or > use solid line
< or < shade below line
> or > shade above line
**Choose a test point to double-check shading; usually (0,0)
Quadratic Inequalities:Example
y<x2+5x+10
Quadratic Inequalities:Example
y>x2-7x-18
Quadratic Inequalities: Worksheet Example
y<-2x2-8x-12
Vertex Form
y=a(x-h)2+k
vertex:(h,k)
AOS:h
Vertex Form:Transform
X+h: translate left h units (x-h)
X-h: translate right h units (x+h)
+k: translate up k units (y+k)
-k: translate down k units (y-k)
|a|>1: stretch vertically by a units (y*a)
|a|<1: compress vertically by a units (y*a)
-a: reflect across x-axis (y*-1)
Vertex Form:Transform Example
y=-0.5(x-2)2+3
➔Reflected over x-axis
➔Compressed by 0.5
➔Right 2
➔Up 3
Vertex Form:Transform Example
Original
Reflected (x)
Compress (.5)
Right (2)
Up (3)
(-2,4)
(-2,-4)
(-2,-2)
(0,-2)
(0,1)
(-1,1)
(-1,-1)
(-1,-0.5)
(1,-0.5)
(1,2.5)
(0,0)**vertex
(0,0)
(0,0)
(2,0)
(2,3)
(1,1)
(1,-1)
(1,-0.5)
(3,-0.5)
(3,2.5)
(2,4)
(2,-4)
(2,-2)
(4,-2)
(4,1)
**Can also plug in x-values into equation to get y-values
and coordinates for graphing
Vertex Form:Transformation Worksheet Example
y = −0.5(x + 1)2 − 7
Vertex Form: To Standard Form
y=a(x-h)2+k
y=a(x-h)(x-h)+k
y=a(x2-hx-hx+hh)+k
y=a(x2-(h+h)x+hh)+k
y=ax2-a(h+h)x+ahh+k
Vertex Form: To Standard Form Example
y=3(x-1)2+5
Word Problem Example
1. You throw a bowling ball off of a cliff into the ocean.
Its height as a function of time could be modeled by the
function h(t)=-16x2+16x+480, where t is the time in seconds
and h is the height in feet.
a. How long did it take for the bowling ball to reach his
maximum height?
b. What was the highest point that the bowling ball reached?
c.The bowling ball hit the water after how many seconds?
Word Problem Worksheet Example
The number of bacteria in a refrigerated food is given by
n(t)= 20t2 - 20t + 120, where T is the temperature of the
food in Celsius and n is the number of bacteria in the food.
At what temperature will the number of bacteria be minimal?