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.

- 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) = ax*is a parabolic curve that always crosses the^{2}+ bx + c*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 = ax*.^{2}+ 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 =
*x*^{2}- 6*x*+ 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.

- Transform the parent quadratic function,
*y = x*^{2}, 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) = ax*.^{2}+ bx + c

- 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 *v*_{0} 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

- 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 = ax*, where^{3}+ bx^{2}+ cx + d*a*≠ 0, - Identify the type of symmetry and relate to odd/even exponents.

Task related to this benchmark: Match That Function

- Key characteristics include domain and range, intercepts, end
behavior, and degree.
Know that polynomial functions of degree n have the general form f(x) = a

_{1}x^{n}+ a_{2}x^{n-1}+ ... + a_{n-2}x^{2}+ a_{n-1}x + a_{n}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.

- 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.

- 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) = ax*(^{n}*a*≠ 0) for negative integral values of*n*.

- Simple rational functions are those with linear, quadratic, or
monomial denominators, including power functions of the form
*f(x) = ax*(^{n}*a*≠ 0) for negative integral values of*n*.

- 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.