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Chapter 4, Part 1
4.1 Quadratics Functions and Transformations
A: Quadratic Functions
A QUADRATIC function is an equation in the form:
y  ax 2  bx  c
The _________ of a quadratic equation is a PARABOLA.
Ex. Circle the parabolas. Cross out the others.
y  2 x 2  15 x  18
y  6 x  9
y  x2  4x  4
y  16 x  22
B: Translations of x2
All parabolas are a translation of the parent graph y  x 2
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We discussed translations in chapter 2, lesson 2.6.
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Use your graphing calculator to graph the following translations and
state the transformations to the parent function.
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Ex. 2. y  3( x 1) 2  5
Ex. 1. y  2( x  3) 2 1
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Chapter 4, Part 1
Ex. 3. Match the graph with the equation
A
B
C
D
C: Vertex Form
Quadratic functions can also be written in “vertex form”:
y  a ( x  h) 2  k
The Vertex = ____________
Domain: the set of all x values for the equation
For an up/down parabola, usually:
The Axis of Symmetry = ____________
______________________________
“a” = _________________ of opening
Range:
positive: opens _______
the set of all y values for the equation
For an up/down parabola, usually:
negative: opens _______
_______________________________
Min/Max: The highest or lowest “y” value on
the graph, the y coordinate of the ___________
Ex. 4 Use your graphing calculator to graph:
y  2( x  2)2  5
State:
Vertex: __________
Min or Max? ______
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Value:_____
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A.O.S.: _________
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Domain: _________________
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Range: ___________________
pg 3
Chapter 4, Part 1
Ex. 5 Use your graphing calculator to graph:
y  3x 2  4
State:
Vertex: __________
Min or Max? ______
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Value:_____
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A.O.S.: _________
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Domain: _________________
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Range: ___________________
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WITHOUT A GRAPHING CALCULATOR:
Use a t-chart to graph the following. State the vertex and the equation of the axis of symmetry.
Ex. 6
y  ( x  5)2  2
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Vertex: _______
Make It Happen:
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A.O.S.: _______
1. Determine the
vertex. Plot the
point.
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2. Determine the A.O.S.
Draw in the vertical
line.
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3. Pick a few x-values
for your t-chart that
are to the right of
the A.OS.
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Ex. 7
y  2( x  4)  6
2
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4. Use your t-chart to
find 2 points on the
graph.
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Vertex: _______
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A.O.S.: _______
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5. Use your A.O.S. to
reflect these points
to the other side.
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6. Sketch in the curve.
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Chapter 4, Part 1
D: Using Vertex Form  Find an Equation from a graph
Recall that vertex form: y  a( x  h)2  k
(h,k) is the vertex
(x,y) is any point on the curve
Example 8: Find the equation that models the graph at the right.
Step 1: What is the VERTEX: ______
so h = _____ and k = ______
Step 2: What is a POINT on the graph?
_______
so x = _____ and y = _______
Step 3: Substitute into vertex form and solve for a.
y  a ( x  h) 2  k
Step 4: Plug a, h, and k back into your vertex form.
Example 9: Find the equation that models the graph
Use these to FIND: ____
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Chapter 4, Part 1
4.2 Standard Form of a Quadratic Function
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A: Standard Form, Finding a Vertex
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If a quadratic function is ALREADY in vertex form y  a( x  h)2  k ,
you can find the vertex just by looking at the equation.
If a quadratic function is in STANDARD FORM y  ax 2  bx  c ,
you have to do a little work first to find the vertex.
Ex. 1 Use a graphing calculator to graph
y   x2  2x  1
Vertex: ______
Min/Max? _______ Value: ______
Ex.
y   x2  2x  1
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A.O.S.: ______
Domain: ________________________________
Finding a vertex
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Range: _________________________________
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Chapter 4, Part 1
Find the vertex, the axis of symmetry, the minimum or maximum value, the domain, and the range.
Ex. 2
y  2 x2  8x  1
Ex. 3
y  3x 2  6 x  9
Ex. 4 WITHOUT A GRAPHING CALCULATOR
Use a t-chart to graph y  x 2  2 x  3
State the vertex and the axis of symmetry
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Chapter 4, Part 1
B: Standard Form  Vertex Form
y  ax 2  bx  c  y  a( x  h)2  k
Ex. 5 Rewrite y   x 2  4 x  5 in vertex form.
Step 1: Identify a & b
Step 2: Find the x coordinate of the vertex.
This value = h
Step 3: Substitute in the x-coordinate to find
the y-coordinate.
This value = k
Step 4: Write out vertex form.
Step 5: Substitute a, h, and k (Simplify if needed)
Rewrite in vertex form
Ex. 6
y  2 x 2  10 x  7
Ex. 8 Which is the graph of y  3x 2  4 x  6 ?
Ex. 7
y  x 2  16 x  66
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Chapter 4, Part 1
4-3 Modeling with Quadratic Functions
A: What is a Model? How do I make one?
A model is a way of using math to
describe a real world situation.
To model a quadratic equation, you
need to know the values of a, b, and c
in y  ax 2  bx  c .
Ex. 1
What is the equation of the parabola that contains the points (0,0), (1,-2), and (-1,-4)?
Make a Plan
 We need a, b, c
 We HAVE three points (x, y)
Make it Happen
 Substitute each of the three points (x,y) into y  ax 2  bx  c
 Solve the 3x3 systems of equations (like section 3.5)
 Substitute a, b, & c into y  ax 2  bx  c
y  ax 2  bx  c
Use (0,0)
Use (1,-2)
Use (-1,-4)
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Chapter 4, Part 1
X
0
2
1
Y
3
5
6
Ex. 2 Write the equation of the quadratic equation that passes through the points on the table.
B: Quadratic Regression
Sometimes your data points will not be EXACTLY a parabola, but the
basic shape of the data is parabolic. We can use a graphing calculator
to ESTIMATE the equation of a parabola that models the data.
Ex. 3 What is the quadratic model for
the data? (start by changing to a
24 hr clock!)
Quadratic Regression in Graphing
Calculator
1. Press the STAT key
2. Choose Option 1 “EDIT”
3. Enter your “x” data in the L1 list
Enter your “y” data in the L2 list
4. Press the STAT key
5. Arrow right to the CALC menu and
choose #5 Quad Reg
6. Enter
7. Use the estimated a, b, &c to write your
equation
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Chapter 4, Part 1
Ex. 4 Use quadratic regression to find a quadratic model for the
data in the chart.
a = _________________________
b = ________________________
c = ________________________
Equation: _______________________________________________
Use the equation to estimate the number of subscribers in 1995.
Ex. 5
Use quadratic regression to find a quadratic model for the data in the chart.
a = _____________________
x=0
b = _____________________
x=10
x=20
c = _____________________
x=29
Equation: ___________________________________________________
Use the equation to estimate the price per gallon in 2006 (x = 30).