Post on 14-Sep-2020
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: π(π₯) = 2π₯ β 6 1. Identify the y-intercept by substituting 0 for x:
( _______ , _______ )
2. Identify the x-intercept substituting 0 for π(π₯):
( _______ , _______ )
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?
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 (π₯ β 4)(π₯ + 8) = 0, find the value(s) for π₯. Show your work and explain how you got your answers.
6. If we are given (π₯ β 4)(π₯ + 8)(π₯ β 2) and their product is 0, then one of the individual
factors MUST be 0. Therefore, (π₯ β 4)(π₯ + 8)(π₯ β 2) = 0 when π₯ = 2, π₯ = _______, πππ π₯ = ______.
7. Solve (π₯ β 10)(π₯ + 6) = 0 π₯ = _______ πππ π₯ = __________
8. Solve (2π₯ β 8)(π₯ β 12) = 0 π₯ = ______ πππ π₯ = __________
9. Solve (3π₯ β 2)(5 β π₯) = 0 π₯ = ______ πππ π₯ = __________
10. Solve (π₯ β π΄)(π₯ β π΅) = 0 π₯ = ______ πππ π₯ = __________ 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 π(π) = (ππ β π)(π β π)
1. Determine the y-intercept by substituting 0 for x: y-intercept: ( _____ , _____ )
2. Determine the x-intercepts by substituting 0 for π(π₯): (Remember our zero-product rule!!) 2π₯ β 6 = 0 πππ π₯ β 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: ______ (letβs call it m)
Explain how you determined this value:
5. Calculate π(π).
Identify the vertex ( π , π(π) )= ____________
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: π(π) = (ππ β π)(π β π)
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)
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. π(π₯) = (π₯ β 1)(π₯ β 3)
y-intercept: (_____ , ______) x-intercepts: (_____ , ______) and ( _____ , ______) vertex: ( _____ , ______)
2. π(π₯) = (π₯ + 1)(π₯ β 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. β(π₯) = (π₯ + 1)(π₯ + 3)
y-intercept: (_____, ______) x-intercepts: (_____, ______) and (_____, ______) vertex: (_____, ______)
4. π(π₯) = β2(π₯ β 1)(π₯ β 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. πΉ(π₯) = (400 β π₯)π₯ vertex is (_____, ______)
2. πΊ(π₯) = β(400 β π₯)(100 β π₯) vertex is ( _____, ______ )
3. π»(π₯) = π₯(π₯ β 8) vertex is ( _____, ______ )
4. π½(π₯) = β(π₯ β 2)(π₯ β 13) vertex is ( _____, ______ )
5. πΎ(π₯) = (π₯ + 5)(π₯ + 9) vertex is ( _____, ______ )
6. πΏ(π₯) = (2π₯ + 6)(3π₯ β 30) vertex is ( _____, ______ )
7. π(π₯) = (240 β 2π₯)(5π₯ + 100) vertex is ( _____, ______ )
8. π(π₯) = (3π₯ β 2)(π₯ + 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:
π(π₯) = (π₯ β π)(π₯ β π)
Identify the x-coordinate of the vertex of the function: ______________
3. Given the x-intercepts of a quadratic function:
)0,34( and )0,34(
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: π(π₯) = 2π₯ β 6 4. Identify the y-intercept by substituting 0 for x:
( _______ , _______ )
5. Identify the x-intercept substituting 0 for π(π₯):
( _______ , _______ )
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?
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 (π₯ β 4)(π₯ + 8) = 0, find the value(s) for π₯. Show your work and explain how you got your answers.
16. If we are given (π₯ β 4)(π₯ + 8)(π₯ β 2) and their product is 0, then one of the individual
factors MUST be 0. Therefore, (π₯ β 4)(π₯ + 8)(π₯ β 2) = 0 when π₯ = 2, π₯ = _______, πππ π₯ = ______.
17. Solve (π₯ β 10)(π₯ + 6) = 0 π₯ = _______ πππ π₯ = __________
18. Solve (2π₯ β 8)(π₯ β 12) = 0 π₯ = ______ πππ π₯ = __________
19. Solve (3π₯ β 2)(5 β π₯) = 0 π₯ = ______ πππ π₯ = __________
20. Solve (π₯ β π΄)(π₯ β π΅) = 0 π₯ = ______ πππ π₯ = __________ 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 π(π) = (ππ β π)(π β π)
7. Determine the y-intercept by substituting 0 for x: y-intercept: ( _____ , _____ )
8. Determine the x-intercepts by substituting 0 for π(π₯): (Remember our zero-product rule!!) 2π₯ β 6 = 0 πππ π₯ β 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: ______ (letβs call it m)
Explain how you determined this value:
11. Calculate π(π).
Identify the vertex ( π , π(π) )= ____________
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: π(π) = (ππ β π)(π β π)
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)
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. π(π₯) = (π₯ β 1)(π₯ β 3)
y-intercept: (_____ , ______) x-intercepts: (_____ , ______) and ( _____ , ______) vertex: ( _____ , ______)
6. π(π₯) = (π₯ + 1)(π₯ β 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. β(π₯) = (π₯ + 1)(π₯ + 3) y-intercept: (_____, ______) x-intercepts: (_____, ______) and (_____, ______) vertex: (_____, ______)
8. π(π₯) = β2(π₯ β 1)(π₯ β 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. πΉ(π₯) = (400 β π₯)π₯ vertex is (_____, ______)
10. πΊ(π₯) = β(400 β π₯)(100 β π₯) vertex is ( _____, ______ )
11. π»(π₯) = π₯(π₯ β 8) vertex is ( _____, ______ )
12. π½(π₯) = β(π₯ β 2)(π₯ β 13) vertex is ( _____, ______ )
13. πΎ(π₯) = (π₯ + 5)(π₯ + 9) vertex is ( _____, ______ )
14. πΏ(π₯) = (2π₯ + 6)(3π₯ β 30) vertex is ( _____, ______ )
15. π(π₯) = (240 β 2π₯)(5π₯ + 100) vertex is ( _____, ______ )
16. π(π₯) = (3π₯ β 2)(π₯ + 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:
π(π₯) = (π₯ β π)(π₯ β π)
Identify the x-coordinate of the vertex of the function: ______________
7. Given the x-intercepts of a quadratic function:
)0,34( and )0,34(
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?