Transcript Ch 9 Alg 1 07-08 CB, CP

Chapter 9
Quadratic Equations
And
Functions
By Chris Posey and Chris Bell
9-1 Square Roots
You already know how to find the square of a
number i.e. 4²=16
To find the square root, you basically do the
opposite of squaring a number. i.e. √(16)=4
Apply Skills Learned 
• Now that you know how find the square
root, let’s try some problems for practice!
• Find the square roots of the following problems.
3. √(400)
3. √400=20
2. √(16)
4 √(36)
4. √36=6
2. √16=4
1. √(144)
1. √122=12
9-2 Solving by Quadratic Equations
by Finding Square Roots
Example 1> x2=4
x= +√4 or - √4
x= 2 or -2
Example 2> 3x2-48=0
3x2=48
x2=16
x= √16
x=4 or -4
Write the original equation
Find square roots
22=4 and (-22)=4
Write the original equation
Add 48 to each side
Divide each side by 3
Find square roots
42=16 and (-4)2=16
Apply Skills Learned 
• Now that you know how to solve quadratic
equations by finding square roots, find the
answers to the following problems.
2. x2=225
3. x2-15=10
3. x=5
2. x=15
1.5t2-125=0
1. t=5
9.3
Simplifying Radicals
• Product Property of Radicals: √ab= √a ∙ √b
Example> √50= √(25∙2)= 5 √2
Example> √48= √(4∙12)= 4 √3
• Quotient Property of Radicals: √a/b= √a/√b
Example> √32/50= √(2∙16)/√(2∙25)
Factor using square
roots
=√(16/25)
=√(16)/√(25)
=4/5
Divide common factors
Use Quotient Property
Simplify
Apply Skills Learned 
Now that you know how to simplify radicals,
try it out on your own!
1.√(9/49)
2. √(18)
3. √(196)
1)3/7 2)3√2 3)14
9.4
Graphing Quadratic Functions
• A quadratic function is
a function that can be
written as a formula
Y=ax2+bx+c, where a≠0
This will give a graph
with a u shape called a
parabola. If a is grater
than 0, it opens up. If it
is negative, then it
opens down.
Graphing (9.4)
• Find the x coordinate of the vertex, which
is x=-b/2a
• Make a x,y table and use the x values
• Plot the points and connect them with a
smooth curve to form the parabola
Example
• Sketch graph of y=x2-2x-3
• Find x coordinate of vertex.
-b/2a=-2/2(1)=1
• Make a table
x| -2
-1
0
1
2
3
4
y|5
0
-3
-4
-3
0
5
(1,-4) is the vertex, plot the rest of the points
and draw a curve connecting them.
Practice
• Find vertex coordinates, and make a x,y
value table using x values to the right and
left of the vertex.
• y=-4x2-4x+8
y|-16 0 8 9 8 0 -16
(-1/2,9) x|-3 -2 -1 -.5 0 1 2
9.5
Solving Quadratic Equations by
Graphing
• Solutions for the quadratic graphs are the
x-axis intercepts, where y=0.
• This number can be checked in the
original equation, by setting it equal to 0.
Example
•y=x2-2x-3 is shown here.
Note that the x intercepts are
located at -1 and 3.
•If substituted for x, the
equation would result in
zero, the solutions for the
equation.
Practice
• Solve the equation algebraically, check
your answers by graphing.
• 2x2+8=16
Solutions are ±2
9.6
Solving Quadratic Equations by the
Quadratic Formula
• The solutions of the quadratic equation,
ax2+by+c=0, are
(-b+/-√(b2-4ac)
x= --------------------(2a)
when a≠0 and b2-4ac≥0
Example
• Solve x2+9x=14=0
• Solution 1x+9x+14=0
(-b+/-√(b2-4ac)
x= --------------------(2a)
(-(9)+/-√((9)2-4(1)(14))
x= ------------------------------(2(1))
-9+/-√(25)
x=-------------------2
-9+/-5
X=---------------2
There are 2 solutions x=-2, and x=-7
Practice
• Solve the quadratic equation.
• 2x2-3x=8
X=2.89, and x=-1.39
9.7
Using the Discriminant
• The discriminant is the radical expression in the
quadratic formula
ie.
(-b+/-√(b2-4ac))
x= --------------------(2a)
If the discriminant is positive, then the solution has
2 solutions.
If it is zero, it has one solution.
If it is negative, there are no real solutions.
Example
• Find value of the discriminat and determine if it
has two solutions, one solution, or no solutions.
• x2-3x-4=0
• Use the equation, ax2+bx+c=0 to identify values,
ie. a=1, b=-3, c=-4
• Substitute into the discriminant
b2-4ac=(-3)2-4(1)(-4)
=9+16
=25
Discriminant is positive, therefore two solutions.
Practice
• Determine whether the graph will intersect
the x-axis at one, two, or zero points.
• y=x2-2x+4
It does not intersect the axis
9.8
Graphing Quadratic Inequalities
• The graph of a
quadratic inequality
consists of all the
points (x,y) that are
part of the inequality.
Quadratic inequalities can be
represented by;
y> or < or  or  ax2+bx+c
• When graphing, use a dashed line for the
parabola when the equality is > or<
• Use a solid line when it is  or 
• The parabola separates the graph into
two sections. A test point is a point that is
not on the graph.
If the test point is a solution, then shade
the region, if not shade the other region.
Example
• Solve –x2 + 4 < 0.
• Find the x-axis intercepts
–x2 + 4 = 0
x2 – 4 = 0
(x + 2)(x – 2) = 0
x = –2 or x = 2
• Use the origional inequality to
find the area to shade
• y<0, therefore shade
everything outside the
parabola, below the x-axis.
Practice
• Determine whether the orderd pair is a
solution of the inequality.
• y≤x2+7, (4,31)
Point is a solution outside of the parabola
Now remember to study hard,
because we’ll be watching you….