CHAPTER 9 REVIEW
Lesson 9.1- Graphing Quadratic Functions
To graph a quadratic function, you must FIRST find the vertex (h, k)!!
(A) If the function starts in standard form y  ax 2  bx  c :
b
1st: The x-coordinate of the vertex, h, =
2a
2nd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
solving for y.
(B) If the function starts in intercept form y  a( x  p)( x  q ) :
1st: Find the x-intercepts by setting the factors with x equal to 0 & solving for x.
2nd: The x-coordinate of the vertex is half way between the x-intercepts.
3rd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
solving for y.
(C) If the function starts in vertex form y  a( x  h)2  k :
1st: pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign
as what is in the parenthesis!!
2nd: Pick out the y-coordinate of the vertex, k. It will have the SAME sign as the what is in the
equation!
AFTER finding the vertex:
 Make a table of values with 5 points: The vertex, plug in 2 x-coordinates SMALLER than
the x-coordinate of the vertex & 2 x-coordinates LARGER than the x-coordinate of the
vertex.

OR use the 1,3,5 short cut to plot the points and reflect those points
Maximum and Minimum - Direction of Opening:
If a is positive, the graph opens up and has a minimum
If a is negative, the graph opens down and has a maximum
1
Graph each function by making a table of values with at least 5 points or use the 1,3,5 shortcut.
(A)Determine whether the function has a maximum or minimum and state it. (B) State the
Domain and Range of the function. (C) State the vertex. (D) Find the equation of the axis of
symmetry.
1.) f ( x) 
1
( x  6) 2  5
2
y
10
Maximum or Minimum ________
9
8
7
6
Domain: __________________
5
4
3
2
Range: __________________
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Vertex: _________
-2
-3
-4
-5
Equation of symmetry______________________
-6
-7
-8
-9
-10
2.) k(x) = x2 + 2x + 1
y
10
9
8
7
6
5
4
Maximum or Minimum ________
3
2
1
Domain: __________________
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
Range: __________________
-4
-5
-6
Vertex: _________
-7
-8
-9
-10
Equation of symmetry______________________
3.) f(x) = x2 – x – 6
y
10
9
8
7
6
Maximum or Minimum ________
5
4
3
Domain: __________________
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Range: __________________
-2
-3
-4
Vertex: _________
-5
-6
-7
-8
-9
Equation of symmetry______________________
-10
2
4.) f ( x)   x2  2 x  3
y
10
9
8
7
Maximum or Minimum ________
6
5
4
3
Domain: __________________
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Range: __________________
-2
-3
-4
Vertex: _________
-5
-6
-7
-8
Equation of symmetry______________________
-9
-10
5.) h( x)  2 x 2  4 x  1
y
10
9
8
7
Maximum or Minimum ________
6
5
4
3
Domain: __________________
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Range: __________________
-2
-3
-4
Vertex: _________
-5
-6
-7
-8
Equation of symmetry______________________
-9
-10
1
6.) g ( x)   ( x  4)( x  6)
3
y
10
9
8
Maximum or Minimum ________
7
6
5
4
Domain: __________________
3
2
1
Range: __________________
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
Vertex: _________
-4
-5
-6
-7
-8
Equation of symmetry______________________
-9
-10
3
Understand how a function shifts and moves in relation to the parent function f(x) = 𝒙𝟐
Lesson 9.3 – Transformations of Quadratic Function
*Be able to describe the transformation given the function
 A translation (left or right) occurs when you add or subtract a constant to the parent function
o g(x) = 𝑥 2 − 4
 A translation (up or down) occurs when you add or subtract a constant from x in the parent
function
o g(x) = (𝑥 + 2)2
 A dilation occurs when |𝑎| > 1 or |𝑎| < 1
o The graph is stretched when |𝑎| > 1
 g(x) = 3𝑥 2
o The graph is compressed when |𝑎| > 1
 g(x) = −2𝑥 2
 A reflection over the x-axis occurs when the value of a is negative
o g(x) = −𝑥 2
Methods for Solving Quadratic Equations:
A.) Unit 5 (chapter 8) – Factoring
1st: Set equal to 0
2nd: Factor out the GCF
3rd: Complete the X & box method to find the factors or use the Boston Method
4th: Set every factor that contains an x in it, equal to 0 & solve for x.
B.) Lesson 9.2 –Solving Quadratic Equations by Graphing
−𝑏
1st: Find the vertex using 𝑥 = 2𝑎
2nd: Graph the vertex and several other points on either side of the axis of symmetry
3rd: Determine the x-intercepts, these are your solutions
(Remember you can have no solution, one solution or 2 solutions.
C.) Lesson 9.4 - Completing the Square
1st: Move the constant (number with no variable) to the right so that all variables are on the left
& all constants are on the right.
2
2nd - Create a perfect square trinomial on the left side by adding  b  to both sides.
2
 
4th: Factor the left side into a square of a sum or difference.
5th: Take the square root of both sides of the equation. REMINDER: Don’t forget the 
6th: Solve for x
D.) Lesson 9.5 - Quadratic Formula
1st: Set the equation equal to 0.
2nd: Find the values of a, b, and c & plug them into the Quadratic Formula:
x
b  b2  4ac
2a
3rd: Simplify the radical as much as possible.
4th: If possible, simplify the numerator into integers.
5th: Divide. REMINDER: If you have 2 terms in the numerator (ex: 4  6 3 ), divide BOTH
2
terms by the number in the denominator (the example would result in 2  3 3 )
4
Examples: Solve each equation by the method stated.
By Factoring:
7.) x2 – 4x – 5 = 0
8.) 3x2  8x  5  0
9.) 2 x 2  x  15
10.) x2 12 x  32  0
11.) 3 x 2  2 x  21
12.) 2x2  4x  30  0
By Graphing:
13.) y  x 2  16 x  2
14.) y  2 x 2  4 x  7
1
15.) y  ( x  4) 2  8
3
y
y
y
10
10
10
9
9
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-7
-8
-8
-8
-9
-9
-9
-10
-10
-10
5
1
2
3
4
5
6
7
8
9 10
By Completing the Square:
16.) x2 – 6x – 11 = 0
17.) 2y2 + 6y – 18 = 0
18.) 2x2 – 3x – 1 = 0
19.) x2 16x  15  0
20.) 5x2  10 x  20  0
1
21.)  x 2  4 x  6  0
2
By Quadratic Formula:
22.) x2 + x = 12
23.) 5x2 – 8x = -3
24.) 2x2 = 4 – 7x
25.)  x2  3x  5  0
26.) 3x2  2 x  x2  5x 1
6