Chapter 5 – Quadratics, Polynomials and Rational Expressions

# Activity 5a – Transformations of Functions and their Graphs

This activity uses a graphing utility to demonstrate how graphs are transformed by translations, reflections and dilations (magnification). The goal is to give students a visual understanding of how a function, such as $f(x) = \sin x$ can be transformed so that one only needs to memorize facts about one parent function to understand the graphs of an entire class of functions. Some of the functions may be unfamiliar to students, which is fine. It is not necessary to know where the parent function comes from, only that the original function can be transformed. It is the comparison of the two related graphs that is the lesson here.

Transformations of Functions and their Graphs

# Activity 5b – Parabolas: General, Standard and Graphical Forms

This activity establishes connections between three forms of a parabola, the graph, the standard form, $y = a(x-h)^2 + k$, and the general form, $y = ax^2 + bx + c$. The connection between general and standard form is established through equating coefficients. The curve sketching of the graph does not include a discussion of the roots, which is covered in a following activity. The graphing focus is on transforming the graph of $y = x^2$.

Parabolas: General, Standard and Graphical Forms

# Activity 5c – Parabolas: Root Form

This activity continues the previous activity by connecting the root form of a parabola $y = a(x-r)(x-s)$ to the graph, the standard form and the general form. In particular, this activity is designed to give students methods of finding the roots other than completing the square and the quadratic formula, which are saved for future activities. The connection to the graph gives a good visual approach to finding roots.

Parabolas: Root Form

# Activity 5d – Completing the Square

This activity builds the completing the square process from the fundamentals. Applications to the equation of a circle and the standard form of a parabola are performed, and therefore it may be useful to review these topics.

Completing the Square

# Activity 5e – The Quadratic Formula, Three Different Ways

This activity connects the quadratic formula to the root, general and standard forms of the parabola. The first method is not a derivation of the quadratic formula, but merely a verification from the root form. The second method is the standard completing the square derivation, beginning from the general form. The third method starts from the standard form to derive a much simpler version of the quadratic formula, then derives the more familiar version by equating coefficients between the general and standard forms and substituting.

The Quadratic Formula, Three Different Ways