3 (January 17 th –
March 23 )
Unit Overview
Unit 6: Quadratics
Another important type of function studied in this course is a quadratic function. A quadratic function is a function defined by an equation of the form y
= ax
2
+ bx + c. This is called the standard form of a quadratic function. Function tables are used to build the basic shape of the graph of quadratic functions and demonstrate that not all graphs are linear. Basic components of the graph are identified, including the vertex, axis of symmetry, and roots.
The difference between the number of roots of a linear function and a quadratic function is identified and the idea of “double root” and “no real roots” are introduced. Simple transformations of the parent graph are shown to illustrate the relationship of the equation and the graph. The factored form of a quadratic function is then introduced as y = a(x-p)(x-q). It is observed that p and q are the roots of the function, and that the x -coordinate of the vertex of a quadratic function lies halfway between the roots of the function. Given a quadratic function in factored form, the roots and vertex are found, and the graph of the function is drawn. Once factored form of the quadratic is introduced, polynomial multiplication is demonstrated as a means to move from factored form to standard form of the quadratic.
The concept of a solution to an equation in one variable is once again linked with the intersections of the graphs. In particular, it is discussed that the solutions to 0 = ax
2
+ bx +c represent the roots of the function y = ax
2
+ bx +c. The quadratic formula is then introduced as a way of determining the solutions to a quadratic equation, or the roots of a quadratic function. It is emphasized that the equation must be written in standard form to use the quadratic formula. The quadratic formula is used to solve a wide variety of quadratic equations, including problems modeling the motion of a launched or falling object. The zero product property is re-introduced and used as a means to find the solutions to quadratic equations that are in factored form.
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Using the found roots of a quadratic function, sketches of the graph of the function are created, and the relationship among solutions, roots, and xintercepts is communicated.
Math Literacy
Key Vocabulary: Quadratic equation, Quadratic formula, parabola, axis of symmetry, vertex, maximum, minimum, double root, discriminant, completing the square, factor, factoring, zeros, roots, completing the square, x-intercepts, complex solutions, zero product property, standard form of a quadratic equation, vertex form.
Supporting Vocabulary: Function, solution, distributive property, square roots, expression, equation, polynomial, binomial, trinomial, degree of a polynomial, leading coefficient, prime polynomial, difference of two squares, perfect square trinomial, degree of a polynomial.
Common Core State Standard Alignment

A.SSE.3a
Factor a quadratic expression to reveal the zeros of the function it defines.

A.SSE.3b
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A.REI.4a
Use the method of completing the square to transform any quadratic equation in x into an equation of the form ( x – p ) 2 = q that has the same solutions. Derive the quadratic formula from this form.

A.REI.4b
Solve quadratic equations by inspection (e.g., for x
2
= 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a
± bi for real numbers a and b .

F.BF3
Identify the effect on the graph of replacing f ( x ) by f ( x ) + k , k f ( x ), f ( kx ), and f ( x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

F.IF.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity .
★

FI.IF.7A
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
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complicated cases.
Graph linear and quadratic functions and show intercepts, maxima, and minima.

FI.IF.8A
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Sample Essential Questions
•
How can you relate representations of quadratic functions, such as algebraic, tabular, and graphical?
•
When could a nonlinear function, like the quadratic function be used to model a real world-situation?
•
How would you decide the best method to solve a quadratic?
•
What are the connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts of the graph of the function?
Anchor Standards/Central Concepts
20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
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California State Standard
14.0 Students solve a quadratic equation by factoring or completing the square.
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions
3

14a I can explain the zero product property and use it to solve quadratic equations in factored form.

14b I can solve a quadratic equation by completing the square to find the roots/zeroes/x-intercepts/solutions of a quadratic and check solution(s).

Where does the graph of: f (x) = 𝑥 3 + 2𝑥 2 𝑥+1
−15𝑥
intersect the x-axis?

What quantity should be added to both sides of this equation to complete the square? x
2
– 14 x = 20 a) -49 b) 49 c) -7 d) 7
4

California State Standard
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions
19.0
Students know the quadratic formula and are familiar with its proof by completing the square.
2

19a I can recite, apply, and explain the use the quadratic formula to find the roots/zeroes/x-intercepts/ solutions of a quadratic.

19b I can prove the quadratic formula by completing the square.

19c I can simplify radicals to help find the approximate and exact solution to a quadratic equation.
20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 3

20a I can find the roots of a quadratic using the quadratic formula.

20b I can find the discriminant of a quadratic equation and use it to explain the differences between real and non-real solutions of quadratics.

20c I can explain the conditions for when a quadratic equation has no real solutions, one real solution, or
Suppose the graph of y = px
2
+ 5 x + 2 intersects the x -axis at two distinct points, where p is a constant. What are the possible values of p ?
Let f( x ) = a x
2
+ b x + c . Suppose that b
2
– 4 ac > 0. Use the quadratic formula to show that it has two roots.
5

California State Standard
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions two real solutions.

20d I can identify the vertex of a quadratic equation when given the roots/solutions/zeroes/x-intercepts of the equation.

20e I can solve quadratic equations by inspection.

20f Given a quadratic equation, I can identify and explain which method should be used to solve.
 20g I can factor to find the roots of a factorable quadratic equation, explain the property being used and why this method is preferable to using the quadratic formula.
Solve for x: 2 x
2
− 3 x
− 5 = 0.
21.0 Students graph quadratic functions and know that their roots are the x-intercepts.
3

21a I can differentiate between the graph of a linear function and the graph of quadratic and
The graph of y = x
( -
1
3
, 0). What is b?
2
+ bx -1 passes through
6

California State Standard
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions explain the definition of a quadratic function.

21b I can find the vertex and axis of symmetry of a quadratic function and explain how these influence the shape of the quadratic.

21c I can solve a quadratic in factored form using Zero
Product Property

21d I can determine the relationship between standard form and factored form.

21e I can graph a quadratic function including roots, vertex, and axis of symmetry.

21f I can identify the relationship among solutions, roots, and x-intercepts of a quadratic function.

21g I can graph the parent function y = x
2
and identify the major components of the graph
What is the x -intercept of the graph of y = x
2
– 4 x + 4 ?
A.
(-2,0)
B.
C.
D.
E.
(-1,0)
(0,0)
(1,0)
(2,0)
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California State Standard
22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
1
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions
(vertex/maximum or minimum, axis of symmetry, roots/solutions/zeroes/xintercepts)

21h I can use function tables to graph quadratic functions and explain the connections to graphing linear functions by the same method.

21i I can determine the number of real solutions when given the graph of a quadratic equation.

21j I can find roots/solutions/zeroes/xintercepts of a quadratic function in factored form, explain each step in the process, and use that information to sketch a graph of the function.

22a I can explain what the discriminant tells me about a quadratic.

22b I can use factoring to determine how many times a
At how many points does the graph of g(x) = 2 x
2
− x + 1 intersect the x -axis?
8

California State Standard
Learning Targets/
Key Concepts and Skills
College-Ready Sample Assessment
Questions graph intersects the x -axis.
23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
3

23a I can translate and solve physical motion problems.

23b I can translate and solve area problems involving quadratics.

23c I can explain why quadratic equations are used to represent the height of launched or falling object.
Vertical Alignment

Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems 7AF 3.1

Use order of operations to evaluate algebraic expressions 7AF 1.2

Solve and graph quadratic equations and quadratic equations in the complex number system 2A8.0

Use the definition of logarithms to translate between logarithms in any base 2A13.0
Additional Sample College-Ready Assessments Questions
A ball is launched from the ground straight up into the air at a rate of 64 feet per second.
Its height h above the ground (in feet) after t seconds is h = 64t – 16t
2
.
How high is the ball after 1 second? When is the ball 64 feet high? For what values of t is h = 0? What events do these represent in the flight of the ball? (adapted from CERT
1997, 21)
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
Sample College-Ready Assessment Questions
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