math 2 (l1-2) a.sse.3, f.if.8 assessment title: factored ......math 2 (l1-2) a.sse.3, f.if.8...

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 1191250 =____________

2. Determine ( 7) (315) (0) (89) =____________

3. Determine (13)(21)(0) = ____________

4. What can you conclude from the example problems above?


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 𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________

10. Solve (𝑥 − 𝐴)(𝑥 − 𝐵) = 0 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.



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) )= ____________



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-value

y-intercept 0

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)



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: (_____, ______)



3. ℎ(𝑥) = (𝑥 + 1)(𝑥 + 3)


4. 𝑘(𝑥) = −2(𝑥 − 1)(𝑥 − 3)




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 ( _____, ______ )



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:

)0,34( and )0,34(


4. Suppose a quadratic function has only one x-intercept, what can you conclude about the vertex?



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 1191250 =____________

12. Determine ( 7) (315) (0) (89) =____________

13. Determine (13)(21)(0) = ____________

14. What can you conclude from the example problems above?



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 𝑥 = ______ 𝑎𝑛𝑑 𝑥 = __________

20. Solve (𝑥 − 𝐴)(𝑥 − 𝐵) = 0 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.



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) )= ____________



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-value

y-intercept 0

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)



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)




7. ℎ(𝑥) = (𝑥 + 1)(𝑥 + 3) y-intercept: (_____, ______) x-intercepts: (_____, ______) and (_____, ______) vertex: (_____, ______)

8. 𝑘(𝑥) = −2(𝑥 − 1)(𝑥 − 3)




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 ( _____, ______ )



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)


7. Given the x-intercepts of a quadratic function:

)0,34( and )0,34(


8. 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 ......math 2 (l1-2) a.sse.3, f.if.8...

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