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CHAPTER 5.1 & 5.2
QUADRATIC FUNCTIONS
QUADRATIC FUNCTION
A QUADRATIC FUNCTION is a function
that can be written in the standard
form:
f(x) = ax2 + bx + c where a≠ 0
GRAPHING QUADRATIC
The graph of a quadratic function is Ushaped and it is called a PARABOLA.
a<0
a>0
PARTS OF A PARABOLA!!!
Vertex: highest or lowest
point on the graph.
2 ways to find Vertex:
1) Calculator:
2nd  CALC
MIN or MAX
2) Algebraically
PARTS OF A PARABOLA!!!
Axis of symmetry:
vertical line that
cuts the parabola
in half
Always x = a
Where a is the x
from the vertex
PARTS OF A PARABOLA!!!
Corresponding
Points: Two points
that are mirror
images of each
other over the axis
of symmetry.
PARTS OF A PARABOLA!!!
Y-intercept:
Where the
parabola crosses
the Y-Axis.
To find:
Look at the table
where x is zero.
PARTS OF A PARABOLA!!!
X- Intercept: The
the parabola
cross the x-axis.
To find:
2nd  CALC
Zero,
Left Bound, Right
Bound
FIND EACH ONE
ON ITS OWN!!
TRY SOME!
Find the vertex and axis of symmetry for each
parabola.
TRY SOME!
Find the Vertex, Axis of Symmetry, X-Int and Yint for each quadratic equation.
1. y = x2 + 2x
2. y = -x2 + 6x + 5
3. y = ¼ (x + 5)2 – 3
TRY SOME!
Identify the vertex of the graphs below, the axis of
symmetry and the points that correspond with points
P and Q.
WRITING QUADRATIC EQUATIONS
We can use systems of equations to
write quadratic equations.
THE CALCULATOR CAN DO IT FOR
YOU!
Find a quadratic equation to model the level of
water in the water tank. How much water is in the
tank after 35 second?
When is it empty?
STAT  ENTER
X-values in L1 and y-values in L2
STAT  CALC
5: QuadReg  ENTER
CHAPTER 5.3
TRANSLATING PARABOLA
STANDARD FORM VERTEX
VERTEX FORM
Graph the following functions. Identify the
vertex of each.
1. y = (x – 2)2
2. y = (x + 3)2 – 1
3. y = -3(x + 2)2 + 4
4. y = 2(x + 3)2 + 1
VERTEX OF VERTEX FORM
The Vertex form of a quadratic equation
is a translation of the parent function
y = x2
VERTEX OF VERTEX FORM
IDENTIFYING THE TRANSLATION
Given the following functions, identify the
vertex and the translation from y = x2
1.y = (x + 4)2 + 7
2
2.y = -(x – 3) + 1
3.y = ½ (x + 1)2
4.y = 3(x – 2)2 – 2
WRITING A QUADRATIC EQUATIONS
TRY ONE!
Write an equations for the following
parabola.
ONE MORE!
Write an equation in vertex form:
Vertex (1,2) and y – intercept of 6
CONVERTING FROM STANDARD TO
VERTEX FORM
Things needed:
Find Vertex using x = -b/2a, and
y = f(-b/2a) This is your h and k.
Then use the the a from standard form.
CONVERTING FROM STANDARD TO
VERTEX
Standard: y = ax2 + bx + c
Things you will need:
a=
and Vertex:
Vertex: y = a(x – h)2 + k
EXAMPLE
Convert from standard form to vertex
form.
y = -3x2 + 12x + 5
EXAMPLE
Convert from standard form to vertex
form.
y = x2 + 2x + 5
TRY SOME!
Convert each quadratic from standard to
vertex form.
1. y = x2 + 6x – 5
2. y = 3x2 – 12x + 7
3. y = -2x2 + 4x – 3
WORD PROBLEMS
WORD PROBLEMS
A ball is thrown in the air. The path of
the ball is represented by the equation
h = -t2 + 8t.
What does the vertex represent?
What does the x-intercept represent?
WORD PROBLEMS
A lighting fixture manufacturer has daily
production costs of C = .25n2 – 10n +
800, where C is the total daily cost in
dollars and n is the number of light
fixture produced. How many fixtures
should be produced to yield minimum
cost.
FACTORING
GCF
One way to factor an expression is to factor
out a GCF or a GREATEST COMMON FACTOR.
EX:
4x2 + 20x – 12
EX:
9n2 – 24n
FACTORS
Factors are numbers or expressions that you
multiply to get another number or expression.
Ex. 3 and 4 are factors of 12 because 3x4 = 12
FACTORS
What are the following expressions factors of?
1. 4 and 5?
3. 4 and (2x + 3)
2. 5 and (x + 10)
4. (x + 3) and (x - 4)
5. (x + 2) and (x + 4) 6. (x – 4) and (x – 5)
TRY SOME!
Factor out the GCF:
a. 9x2 +3x – 18
b. 7p2 + 21
c. 4w2 + 2w
FINDING FACTORS OF QUADRATIC
EXPRESSIONS
When a = 1:
x2 + bx + c
Step 1. Determine the signs of the factors
Step 2. Find 2 numbers that’s product is
c, and who’s sum is b.
EXAMPLES
Factor:
1. X2 + 6x + 9
2. x2 – 13x + 42
3. x2 – 5x – 66
4. x2 – 16
Factoring a trinomial:
2
ax  bx  c
1. Write two sets of parentheses,
( )( ). These will be the factors
of the trinomial.
2. Product of first terms of both
binomials must equal first term
2
of the trinomial (ax ) .
Next
Factoring a trinomial:
2
ax  bx  c
3. The product of last terms of
both binomials must equal last
term of the trinomial (c).
4. Think of the FOIL method of
multiplying binomials, the sum
of the Outer and the Inner
products must equal the
middle term (bx).
EXAMPLE!
Factor:
1.
3x2 – 16x + 5
EXAMPLE!
Factor:
2.
2x2 + 11x + 12
EXAMPLE!
Factor:
3.
2x2 + 7x – 9
TRY SOME!
Factor
1. 5t2 + 28t + 32
2. 2m2 – 11m + 15