Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Learning Targets:
 Use factored form to identify key features of a quadratic function
Part I: Looking back at linear functions.
Consider the function: 𝑓(𝑥) = 2𝑥 – 6
1. Identify the y-intercept by substituting 0 for x:
( _______ , _______ )
2. Identify the x-intercept substituting 0 for 𝑓(𝑥):
( _______ , _______ )
3. Using the intercepts, graph the function.
Calculating the x and y-intercepts of a function is a slick way to graph the function. We can use
this technique for other functions that are not linear, such as quadratic functions.
Part II: The Zero Product Rule
1. Determine 0  5  12  119 =____________
2. Determine (7)  (315)  (0)  (89) =____________
3. Determine (13)(21)(0) = ____________
4. What can you conclude from the example problems above?
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
5. If (𝑥 − 4)(𝑥 + 8) = 0, find the value(s) for 𝑥. Show your work and explain how you got
your answers.
6. If we are given (𝑥 − 4)(𝑥 + 8)(𝑥 − 2) and their product is 0, then one of the individual
factors MUST be 0.
Therefore, (𝑥 − 4)(𝑥 + 8)(𝑥 − 2) = 0 when 𝑥 = 2, 𝑥 = _______, 𝑎𝑛𝑑 𝑥 = ______.
7. Solve (𝑥 − 10)(𝑥 + 6) = 0 𝑥 = _______ 𝑎𝑛𝑑 𝑥 = __________
8. Solve (2𝑥 − 8)(𝑥 − 12) = 0 𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
9. Solve (3𝑥 − 2)(5 − 𝑥) = 0
10. Solve (𝑥 − 𝐴)(𝑥 − 𝐵) = 0
𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
In general, to determine when a product of linear factors is equal to 0, just set each individual
factor equal to 0 and solve. This zero-product rule will make our work with quadratic functions
much easier!
When a function is expressed in factored form (written as a product of linear factors), we can,
by the zero-product rule, determine its x-intercepts simply by setting each factor equal to 0.
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part III: Identify the key features and graph the quadratic function 𝒒(𝒙) = (𝟐𝒙 − 𝟔)(𝒙 − 𝟕)
1. Determine the y-intercept by substituting 0 for x:
y-intercept: ( _____ , _____ )
2. Determine the x-intercepts by substituting 0 for 𝑞(𝑥):
(Remember our zero-product rule!!)
2𝑥 − 6 = 0 𝑎𝑛𝑑 𝑥 − 7 = 0
x-intercepts: ( ______ , ______ ) and ( _____ , ______ )
3. The symmetry of parabolas allows us to use the x-intercepts of a quadratic function to
determine its vertex.
In general, if a quadratic function has two x-intercepts, the x-coordinate of its vertex
must be midway between the two x-intercepts:
4. Using the intercepts you found above, determine the value that is midway between:
______ (let’s call it m)
Explain how you determined this value:
5. Calculate 𝑞(𝑚).
Identify the vertex ( 𝑚 , 𝑞(𝑚) )= ____________
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
6. Using the four key points you determined from #1-5,
complete the table of values below and graph the
function: 𝒒(𝒙) = (𝟐𝒙 − 𝟔)(𝒙 − 𝟕)
Key Point
x-value
y-intercept
0
y-value
x-intercept
0
x-intercept
0
vertex
5
In general, when a quadratic function is presented in factored form, you can easily determine
the following to graph the function:
a. y-intercept
b. x-intercepts
c. vertex (using the point midway between the x-intercepts)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part IV: You try it now!
Determine the key points for each function below. Then, use those four key points to graph
the function.
1. 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 3)
y-intercept: (_____ , ______)
x-intercepts: (_____ , ______) and ( _____ , ______)
vertex: ( _____ , ______)
2. 𝑔(𝑥) = (𝑥 + 1)(𝑥 − 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
3. ℎ(𝑥) = (𝑥 + 1)(𝑥 + 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
4. 𝑘(𝑥) = −2(𝑥 − 1)(𝑥 − 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part V: Determine the vertex for each of the following quadratic functions.
1. 𝐹(𝑥) = (400 – 𝑥)𝑥
vertex is (_____, ______)
2. 𝐺(𝑥) = −(400 − 𝑥)(100 – 𝑥)
vertex is ( _____, ______ )
3. 𝐻(𝑥) = 𝑥(𝑥 − 8)
vertex is ( _____, ______ )
4. 𝐽(𝑥) = −(𝑥 − 2)(𝑥 − 13)
vertex is ( _____, ______ )
5. 𝐾(𝑥) = (𝑥 + 5)(𝑥 + 9)
vertex is ( _____, ______ )
6. 𝐿(𝑥) = (2𝑥 + 6)(3𝑥 − 30)
vertex is ( _____, ______ )
7. 𝑀(𝑥) = (240 − 2𝑥)(5𝑥 + 100)
vertex is ( _____, ______ )
8. 𝑁(𝑥) = (3𝑥 − 2)(𝑥 + 7)
vertex is ( _____, ______ )
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part VI - Summary: When working with quadratic functions and graphing parabolas, it is
often important to determine the y-intercepts, the x-intercepts, and the vertex.
1. In general if a quadratic function is presented in factored form, explain how to
determine the vertex:
2. If you are given a general quadratic function in factored form as:
𝑓(𝑥) = (𝑥 − 𝑎)(𝑥 − 𝑏)
Identify the x-coordinate of the vertex of the function: ______________
3. Given the x-intercepts of a quadratic function:
(4  3, 0) and (4  3, 0)
Identify the x-coordinate of the vertex of the function: ______________
4. Suppose a quadratic function has only one x-intercept, what can you conclude about the
vertex?
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Answer Key
Part I: Looking back at linear functions.
Consider the function: 𝑓(𝑥) = 2𝑥 – 6
4. Identify the y-intercept by substituting 0 for x:
( _______ , _______ )
5. Identify the x-intercept substituting 0 for 𝑓(𝑥):
( _______ , _______ )
6. Using the intercepts, graph the function.
Calculating the x and y-intercepts of a function is a slick way to graph the function. We can use
this technique for other functions that are not linear, such as quadratic functions.
Part II: The Zero Product Rule
11. Determine 0  5  12  119 =____________
12. Determine (7)  (315)  (0)  (89) =____________
13. Determine (13)(21)(0) = ____________
14. What can you conclude from the example problems above?
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
15. If (𝑥 − 4)(𝑥 + 8) = 0, find the value(s) for 𝑥. Show your work and explain how you got
your answers.
16. If we are given (𝑥 − 4)(𝑥 + 8)(𝑥 − 2) and their product is 0, then one of the individual
factors MUST be 0.
Therefore, (𝑥 − 4)(𝑥 + 8)(𝑥 − 2) = 0 when 𝑥 = 2, 𝑥 = _______, 𝑎𝑛𝑑 𝑥 = ______.
17. Solve (𝑥 − 10)(𝑥 + 6) = 0 𝑥 = _______ 𝑎𝑛𝑑 𝑥 = __________
18. Solve (2𝑥 − 8)(𝑥 − 12) = 0 𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
19. Solve (3𝑥 − 2)(5 − 𝑥) = 0
20. Solve (𝑥 − 𝐴)(𝑥 − 𝐵) = 0
𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________
In general, to determine when a product of linear factors is equal to 0, just set each individual
factor equal to 0 and solve. This zero-product rule will make our work with quadratic functions
much easier!
When a function is expressed in factored form (written as a product of linear factors), we can,
by the zero-product rule, determine its x-intercepts simply by setting each factor equal to 0.
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part III: Identify the key features and graph the quadratic function 𝒒(𝒙) = (𝟐𝒙 − 𝟔)(𝒙 − 𝟕)
7. Determine the y-intercept by substituting 0 for x:
y-intercept: ( _____ , _____ )
8. Determine the x-intercepts by substituting 0 for 𝑞(𝑥):
(Remember our zero-product rule!!)
2𝑥 − 6 = 0 𝑎𝑛𝑑 𝑥 − 7 = 0
x-intercepts: ( ______ , ______ ) and ( _____ , ______ )
9. The symmetry of parabolas allows us to use the x-intercepts of a quadratic function to
determine its vertex.
In general, if a quadratic function has two x-intercepts, the x-coordinate of its vertex
must be midway between the two x-intercepts:
10. Using the intercepts you found above, determine the value that is midway between:
______ (let’s call it m)
Explain how you determined this value:
11. Calculate 𝑞(𝑚).
Identify the vertex ( 𝑚 , 𝑞(𝑚) )= ____________
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
12. Using the four key points you determined from #1-5,
complete the table of values below and graph the
function: 𝒒(𝒙) = (𝟐𝒙 − 𝟔)(𝒙 − 𝟕)
Key Point
x-value
y-intercept
0
y-value
x-intercept
0
x-intercept
0
vertex
5
In general, when a quadratic function is presented in factored form, you can easily determine
the following to graph the function:
d. y-intercept
e. x-intercepts
f. vertex (using the point midway between the x-intercepts)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part IV: You try it now!
Determine the key points for each function below. Then, use those four key points to graph
the function.
5. 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 3)
y-intercept: (_____ , ______)
x-intercepts: (_____ , ______) and ( _____ , ______)
vertex: ( _____ , ______)
6. 𝑔(𝑥) = (𝑥 + 1)(𝑥 − 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
7. ℎ(𝑥) = (𝑥 + 1)(𝑥 + 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
8. 𝑘(𝑥) = −2(𝑥 − 1)(𝑥 − 3)
y-intercept: (_____, ______)
x-intercepts: (_____, ______) and (_____, ______)
vertex: (_____, ______)
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part V: Determine the vertex for each of the following quadratic functions.
9. 𝐹(𝑥) = (400 – 𝑥)𝑥
vertex is (_____, ______)
10. 𝐺(𝑥) = −(400 − 𝑥)(100 – 𝑥)
vertex is ( _____, ______ )
11. 𝐻(𝑥) = 𝑥(𝑥 − 8)
vertex is ( _____, ______ )
12. 𝐽(𝑥) = −(𝑥 − 2)(𝑥 − 13)
vertex is ( _____, ______ )
13. 𝐾(𝑥) = (𝑥 + 5)(𝑥 + 9)
vertex is ( _____, ______ )
14. 𝐿(𝑥) = (2𝑥 + 6)(3𝑥 − 30)
vertex is ( _____, ______ )
15. 𝑀(𝑥) = (240 − 2𝑥)(5𝑥 + 100)
vertex is ( _____, ______ )
16. 𝑁(𝑥) = (3𝑥 − 2)(𝑥 + 7)
vertex is ( _____, ______ )
Math 2 (L1-2)
A.SSE.3, F.IF.8
Assessment Title: Factored Form of a Quadratic
Unit 3: Quadratic Functions: Working with Equations
Part VI - Summary: When working with quadratic functions and graphing parabolas, it is
often important to determine the y-intercepts, the x-intercepts, and the vertex.
5. In general if a quadratic function is presented in factored form, explain how to
determine the vertex:
6. If you are given a general quadratic function in factored form as:
𝑓(𝑥) = (𝑥 − 𝑎)(𝑥 − 𝑏)
Identify the x-coordinate of the vertex of the function: ______________
7. Given the x-intercepts of a quadratic function:
(4  3, 0) and (4  3, 0)
Identify the x-coordinate of the vertex of the function: ______________
8. Suppose a quadratic function has only one x-intercept, what can you conclude about the
vertex?