Unit 5 Quadratics
Quadratic Functions
• Any function that can be written in the form
Quadratic Functions
• Graph forms a parabola
or
• Label the parts of the parabola
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
•
•
1.
2.
3.
4.
5.
6.
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:
Move the constant
Factor out the # in front of x2
Take ½ of middle term and square it
Write in factored form for the perfect sq.
trinomial
Add to both sides (multiply by # in front)
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 parabola
2. 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
1. 2 x  3 x  10  0
2
2.  x  6 x  15
2
3. x  12  3
2
4.  x  6 x  9
2
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
2 x  7 x  15
2
x  8 x  16
2
3x  5 x  2
2
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
 b  b  4ac 
x
2a
2
Examples
x  12 x  28
2
Examples
x  121  22 x
2
Examples
2x  4x  2  3
2
Examples
x  4 x  13
2
Discriminant
• Used to identify the “type” of solutions you
will have (without having to solve)
b 2  4ac
* * * note that there is no radical over the number
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
ex.  5 x  8 x  1  0
2
ex.  7 x  15 x  4  0
2
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
ex. 4 x  16  0
2
ex.  5x  8
2
Examples
ex. 3x 2  5  6
ex. 16  2  x 2
Examples
ex. 2 x  4 x  3  0
2
ex. 3x  5x  2  0
2
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
2
2
y


2
x
 3x  5
• Ex. y  x  2 x  1
Ex.
•
Graphing Quadratic Inequalities
y  x  4x  5
2
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  2x  8  0
2
Solutions:_______________________
2x  4x  5
2
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 Algebraically
x 2  11x  30  0
x 2  3x  18
4 x  12 x  10
2
Solving Quadratic Inequalities
 4  2 x  8x  2
2
Word Problems