Download Unit 3 Items to Support Formative Assessment

Algebra II Items to Support Formative Assessment
Unit 3: Expanding Understanding of Quadratic Functions
Use complex numbers in polynomial identities and equations.
N.CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 +
4 as (x + 2i)(x – 2i). (Algebra II GT only)
N.CN.C.8 Item 1
Find the value of a, b and c that will make the equation cx2+16=(ax+b)(3x-b) true.
Answer:
a = 3, b = 4i, c=9
N.CN.C.8 Item 2
For each expression select all appropriate factors.
a. x2 + 6ix – 9
b. x2 + 9
c. x2 – 9
d. x2 + 3x – 3ix – 9i
Factors
(x – 3)
(x + 3)
Answers:
a. (x + 3i)
b. (x – 3i)
c. (x – 3)
d. (x + 3)
(x + 3i)
(x + 3)
(x – 3i)
(x – 3i)
(x + 3i)
N.CN.C.8 Item 3
Solve the quadratic equation x2 + 9 = 0 using the quadratic formula. Then, using its solutions,
rewrite the equation to its factored form.
Answer:
x = ±3i
(x + 3i)(x - 3i) = 0
N.CN.C.8 Item 4
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Solve the quadratic equation 4x 2 - 20ix - 25 = 0 using the quadratic formula. Then, using its
solutions, rewrite the equation in its factored form.
Answer:
5i
2
(2x + 5i)(2x + 5i) = 0
x=
Use complex numbers in polynomial identities and equations.
N.CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials. (Algebra II GT only)
N.CN.C.9 Item 1 (Non-calculator)
Consider the function f(x) = 2x2 + 3x + 4
a. How many complex solutions does the function f(x) have? Explain your reasoning.
b. Find all the solutions to the function.
Answer:
a. The function has two complex solutions, according to the Fundamental Theorem of Algebra.
The function has a degree of 2, meaning that it will have exactly two complex solutions.
b. x =
-3 ± i 23
; Though they are imaginary, the function has two complex solutions.
4
N.CN.C.9 Item 2
Part A
Determine if the following statement is true of false. Explain your reasoning.
The quadratic function represented by the graph below has only 1 complex solution.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Part B
Write quadratic equation represented in the above graph in standard form.
Answer:
Part A
False. The graph of the function is a parabola which means its degree is 2. By the FTA, quadratic
functions have exactly 2 complex solutions. The solution x=2 has a multiplicity of 2.
Part B
f(x) = x2 - 4x + 4
N.CN.C.9 Item 3
Consider the following graph for the quadratic function f(x).
Determine if the following statement is true or false. Justify your reasoning.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
This quadratic function has no complex solutions.
Answer:
False. Even though the graph does not have any real zeros, the function will have two complex
solutions. The function has a degree of two (since it is a quadratic function), and according to the
Fundamental Theorem of Algebra, it will have two solutions. These solutions will be imaginary.
N.CN.C.9 Item 4
3
2
Joey solved the quadratic equation 2x 2 - 5x + 3 = 0 and found the solution x = . Here is Joey’s
work.
2x 2 - 5x + 3 = 0
x=
5 ± 25 - 4(2)(3)
4
5± 1
4
5 +1 3
x=
=
4
2
x=
Is Joey’s solution correct? Explain your reasoning.
Answer:
Joey’s solution is not correct. According to the Fundamental Theorem of Algebra, Joey should
have two solutions, since it is a quadratic equation. In his work, Joey found the solution when
adding the terms in the numerator, but did not find the solution for when subtracting the terms in
the numerator. Joey should have another solution of x = -1.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.