Unit 2: Quadratics
What you should already know?
- Definitions: term, factor, coefficient
- Properties of exponents
- Solving equations
- Definition of a zero
- Writing linear and quadratic functions
- Function notation
- Basic properties of numbers (for use in basic proofs/justification)
- Taking the square root
- Simplifying radicals (for the quadratic formula)
- Understanding of equivalency
Interpret expressions that represent a quantity in terms of its context.
Interpret parts of an expression, such as terms, factors, and coefficients.
- I can interpret expressions that represent a context using its context.
- I can interpret terms, factors, and coefficients using a context.
Use the structure of an expression to identify ways to rewrite it. For example,
see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be
factored as (x2 – y2)(x2 + y2).
- I can rewrite expressions using exponent structures.
- I can factor special case expressions based on their structure.
Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the polynomial.
- I can factor polynomials using various methods.
- I can use factors to find zeros of polynomials.
- I can construct graphs of polynomials from their factored forms.
Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions.
- I can write equations and inequalities for a given context.
- I can solve problems using linear and quadratic equations and inequalities.
Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and scales.
- I can write equations in two or more variables to represent a problem.
- I can graph equations on the coordinate plane using proper scales and labels.
Explain each step in solving a simple equation. Note: Limit to factorable
- I can explain each step when solving a simple equation.
Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x2= 49), taking square
roots, the quadratic formula and factoring (when lead coefficient is 1), Writing
complex solutions is not expected; however recognizing when the formula
generates non-real solutions is.
- I can solve quadratic equations by taking the square root.
- I can solve quadratic equations using the Quadratic Formula.
- I can solve quadratic equations by factoring (when a=1).
- I can recognize and explain a complex root.
Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- I can explain why the graph of a variable is the set of all solutions plotted on a plane.
Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
- I can use function notation to describe a quadratic model.
- I can evaluate quadratic functions for inputs in their domains.
- I can interpret quadratic functions based on their context.
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.
- I can interpret key features of a quadratic using its context.
- I can sketch graphs using key features of a quadratic relationship.
Write a function defined by an expression in different but equivalent forms
(including recognizing vertex form) to reveal and explain different properties of the
a. Use the process of factoring to show zeros, extreme values, and symmetry of
the graph, and interpret these in terms of a context.
Note: At this level, completing the square is still not expected.
- I can write a quadratic in all 3 forms: standard, vertex, factored.
- I can use all forms of quadratics to explain key properties.
- I can use factoring to show zeros, extreme values, and symmetry.
- I can interpret the key features in terms of a context.
Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
- I can compare properties of two quadratics given in different forms.
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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.
- I can identify the effect of a constant on a quadratic.