Algebra II End-of-Course Exam Content Standards—Core: Polynommial and Rational Functions (Priority: 30%)
Successful students will be able to use tables, graphs, verbal statements and symbols to represent and analyze quadratic, rational, and higher order polynomial functions. They will be able to recognize and solve problems that can be modeled using these functions. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems.
P1. Quadratic functions
a. Determine key characteristics of quadratic functions and their graphs.
- Key characteristics include domain and range, vertex, minimum/maximum, intercepts, axis of symmetry, and end behavior.
- Recognize that except when a = 0, the graph of f(x) = ax2 + bx + c is a parabolic curve that always crosses the y-axis but may or may not cross the x-axis; that it opens up when a > 0 and down when a < 0.
- Recognize the relationship between the intercepts and the factors of a quadratic function.
- Recognize
as a way of determining the x-coordinate of the vertex for a parabola in the form y = ax2 + bx + c. - Recognize the relationship between complex solutions of a quadratic equation and the characteristics and position of its graph. (i.e. If a graph has no x-intercepts, then the solutions are complex.)
Example: Use different forms of the function to extract information:
- y = x2 - 6x + 8 makes the y-intercept obvious,
- y = (x - 2)(x - 4) provides access to the zeros, and
- y - (x - 3)2 - 1 makes it easy to find the vertex and sketch the graph.
b. Represent quadratic functions using tables, graphs, verbal statements, and equations. Translate among these representations.
c. Describe the effect that changes in the parameters of a quadratic function have on the shape and position of its graph.
- Transform the parent quadratic function, y = x2, to translate, reflect, rotate, stretch or compress the graph.
- Investigate the changes that occur when the coefficients, including the constant are changed in a quadratic function in the form, f(x) = ax2 + bx + c.
d. Recognize, express, and solve problems that can be modeled using quadratic functions. Interpret their solutions in terms of the context.
- Recognize situations, in which quadratic models are appropriate; create and interpret quadratic models to answer questions about those situations.
- This includes using and interpreting appropriate units of measurement and precision for the given application.
Example: Determine the height of an object above the ground t seconds after it has been thrown upward at an initial velocity of v0 feet per second from a platform d feet above the ground.
Example: Determine the relationship between the length of a side of a cube and its surface area.
Tasks related to this benchmark: Leo's Painting, Match That Function, Season Pass
P2. Higher-order polynomial and rational functions
a. Determine key characteristics of power functions in the form f(x) = axn, a ≠ 0, for positive integral values of n and their graphs.
- Recognize the basic power functions as the parent functions for
many types of polynomials.
Example: Determine the parent function for all cubic equations in the form y = ax3 + bx2 + cx + d, where a ≠ 0,
- Identify the type of symmetry and relate to odd/even exponents.
Task related to this benchmark: Match That Function
b. Determine key characteristics of polynomial functions and their graphs.
- Key characteristics include domain and range, intercepts, end
behavior, and degree.
Know that polynomial functions of degree n have the general form f(x) = a1xn + a2xn-1 + ... + an-2x2 + an-1x + an where n is a positive integer. The degree of the polynomial function is the largest power of its terms for which the coefficient is non-zero. The leading coefficient is the coefficient of the term of highest degree.
- Identify and use the degree of a polynomial or its factors to interpret characteristics of the function or its graph.
- Note that every polynomial function of odd degree has at least one real zero.
- Understand the concept of the multiplicity of a root of a polynomial equation and its relationship to the graph of the related polynomial function.
c. Represent polynomial functions using tables, graphs, verbal statements, and equations. Translate among these representations.
- Identify or write a polynomial function of a given degree.
- Decide if a given graph or table of values suggests a higher order polynomial function.
d. Determine key characteristics of simple rational functions and their graphs.
- Key characteristics include domain and range, intercepts, types of symmetry, horizontal and vertical asymptotes, and end behavior.
- Simple rational functions are those with linear, quadratic, or monomial denominators, including power functions of the form f(x) = axn (a ≠ 0) for negative integral values of n.
e. Represent simple rational functions using tables, graphs, verbal statements, and equations. Translate among these representations.
- Simple rational functions are those with linear, quadratic, or monomial denominators, including power functions of the form f(x) = axn (a ≠ 0) for negative integral values of n.
f. Recognize, express, and solve problems that can be modeled using polynomial and simple rational functions. Interpret their solutions in terms of the context.
- This includes using and interpreting appropriate units of measurement and precision for the given application.
Example: The volume of a cylinder whose radius is 5 more than twice the radius, r, of a given sphere and whose height is twice its own radius is 6750π. Determine the radius of the given sphere.