Quadratic Notesb
Anchor Rule Walk (ARW) form:
Rule
y  A  R( x  W ) 2
10
8
y-part
x-part
6
of the VERTEX
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Shape: Every quadratic forms the shape of a parabola, which can look like the letter “n” or “u”.
10
Example:
2
8
OR
-10
-5
5
6
-2
4
-4
2
-6
-5
5
10
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Vertex: is the highest or lowest point of the parabola (maximum or minimum).
-2
How do you identify the Vertex in a table?
-Look only at the y values on the table, then watch for the point that changes the
direction of the y’s.
-10
-4
Examples:
-6
X
5
6
7
8
Y
12
13
12
9
-8
Vertex
-10
X
-15
-14
-13
-12
f(x)
25
22
21
22
Vertex
X
32
33
34
35
Y
2
0
2
8
Vertex
*** Notice the same number is above and below the y-part of the vertex.
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Other Vocabulary:
Y-intercept: The value of y when x is 0. OR
Where the graph crosses the y-axis. OR
Often referred to in word problems as the starting height of an object.
X-intercept: The value of y when x is 0. OR
Where the graph crosses the x-axis. OR
Often referred to in word problems as when the object is on the ground.
Vertex: is also often referred to in word problems as the maximum height.
Meaningful Domain: The positive x-values (or time) that something is in the air. We
ignore the negative x values and negative y values: because x is usually “time”, we
don’t care about negative values (what is negative time?), and y is height and the
object in question probably won’t go underground (normally).
*** Remember that when dealing with word problems you need to understand the above
parts in order to understand what you are given, or what you are figuring out.
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Rule: Given the Vertex and the “next points” you can find rule by answering the
question: How do you get from the y-part of the vertex to the next y?
****The two points can be from a table, a graph or given as coordinates. Suggestion: first put them in a table
Example:
x
2
3
4
5
6
X
3
4
5
y
Y
3
12
15
12
3
To find the rule you must:
1st: Find the vertex (specifically the y-coordinate)
Vertex
2nd: Find the story of the squares! In this case, to
get from 15 to the next y-value, 12, we must
subtract 3. 3= 3*12. To get from 15 to the y-value
after 12, the 3, we subtract 12. 12 = 3*22, and so
on. Therefore, we’re subtracting 3*our square
numbers, so the RULE , R= –3.
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How can you tell the difference between Linear, Exponential, and Quadratic?
It’s all about the Rule
Linear: is repeated addition. So on a table, your rule Y (see linear notes), will be the same
X
for any 2 points you pick. Be sure to try it at least twice. Remember, it is a straight line.
Exponential: is repeated multiplication. On a table, your rule
Y2
(see Exponential notes),
Y1
will be the same for any 2 consecutive points you pick. Be sure you try it at least twice.
For graphing if the rule is: (A) >1 it is Exponential Growth, (B) <1 it is Exponential Decay
Quadratic: Find the vertex, then determine the change in the y coordinate to the “next point”.
Remember, you’re looking for the story of the squares in your y values.
** Remember you need a Rule and a point (or 2 points) to write equations for each. For
quadratic functions, one of those points must be the vertex!
What about in word problems?
1) Linear problems add or subtract a value each time.
2) Exponential problems multiply (ex. Double) or divide (ex. Half) by a value each time, or
percentage changes (e.g. earning interest at a bank).
3) Quadratic problems often have something flying through the air or falling.
** It is important to first identify what type of function you are dealing with before you
can even start working on the problem or writing an equation.
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What does “Translated” mean? In this chapter it means moving the vertex.
Example: If the graph
is translated so that the vertex is now at (-2, 5), write the equation of
the new graph.
Original graph of y = x2,
which a has a vertex of
(0,0)
y  0  ( x  0) 2
y  5  ( x  2)2
Now equation with new
vertex of (-2, 5)
OR
y  5  ( x  2) 2
2
Table to Equation:
Example:
x
-2
-1
0
1
2
3
y
38
24
14
8
6
8
Equation: y =A+R(x - W)2
Vertex = (2, 6), R = +2
Vertex
y  6  2( x  2) 2
To get from 6 to 8 I need
to add 2, So R = +2
(1) If x is 12, what’s y?
(2) If y is 206, what’s x?
y  6  2( x  2) 2 , x  12
****** USE YOUR CALCULATOR ******
y  6  2(12  2) 2
Push
Y=
then type in equation
y  6  2(10) 2
y  6  2(100)
y  6  200
y  206
Push
2nd
then
GRAPH
Now move up and down until you see y = 206.
You should notice that x = – 8 when y = 206
When y = 206, x = – 8
OR
f(– 8) = 206
When x = 12, y = 206
OR
f(12) = 206
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Graphing Calculator Reminder: Be sure to “set up the table” before you view the table on
the calculator.
1st: Push
2nd
then
Window
2nd: adjust TbleStart: The number here will determine where the first x value in the table
will start. This may speed up your search if you pick a good starting point.
3rd: adjust  Tbl: The number here will determine what the Xs will count by. Remember
that you can use whole numbers or decimals here. This can help you
fine tune where a specific Y is located.
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Graph to Equation:
Given a graph, first make a table:
1st: Find the vertex and write it in a table.
2nd: Move in the positive x direction one space, and find the y that goes with it. Put that
point in the table.
rd
3 : Follow the steps above for going from a Table to an Equation.
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Matching Equations to Graphs: Whenever you have to match an equation to a graph
MAKE A TABLE!!!
1st: Write down some points you can read from the graph, in a table
2nd: Plug in the Xs from those points into your equation to see if you get the same Ys.
- If they match, they match.
- If they don’t, they DON’T MATCH.
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3
Solving: Use reverse order of operations so you can get the variable (letter) by itself.
Example:
32  3( x  2) 2  224
Subtract 32 from both sides
3( x  2) 2  192
Divide 3 from both sides
( x  2) 2  64
x  2  8
Square root both sides
x + 2 = 8 and x + 2 = – 8
x=6
and x = -10
Remember there is a positive
and negative root (answer).
There are now 2 equations.
P
E
M/D
A/S
For each equation, subtract 2
from both sides
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ARW to Standard Form: (Standard form is y  ax 2  bx  c )
Example:
y  4( x  5)  1
y  4( x  5)( x  5)  1
First, rewrite the squared term as multiplication.
2
Use Sneaky Squares to Expand (see below)
y  4( x 2  10 x  25)  1
Distribute the 4 to everything inside
the parentheses. Yes, EVERY term.
y  4 x  40 x  100  1
2
y  4 x 2  40 x  101
Combine like terms as the last step.
Sneaky Squares (a.k.a. “the box”)
( x  5)( x  5)
x
x
x2
-5
-5x
Multiply to find each term in each box.
-5
-5x
25
x 2  5 x  5 x  25
Write down each term from each box
x 2  10 x  25
Combine like terms
4