Warm Ups!

Homework 2.1: (in the Big Ideas text book) p. 52-53 #3, 5-6, 9-12, 17, 19-20, 23-25, 27-28, 35-40.

Lesson Goal: At the end of this lesson, you will be able to

identify and define quadratic functions in three forms,
recognize and describe the salient (important) features of a quadratic equation, and
be able to connect the key features of the Vertex form of a quadratic equation with the key features of the graph
understand how changes in the Vertex form of a quadratic equation correspond to transformations of the graph

I will explain these in class, but everything you need to know is summarized here.

Definition and examples: a quadratic function is a polynomial function that can be written in its standard form as

$LaTeX: y\:=\:ax^2+bx\:+\:c\:,\:\:where\:a,\:b,\:and\:c\:are\:real\:numbers\:and\:a\ne0.$

Notice that this definition implies

the highest degree of a quadratic function is 2 and
the coefficients must be real numbers.

Quadratic functions produce U-shaped graphs that are called parabolas.

The parent quadratic function is LaTeX: y=x^2

Notice that in the parent function: $LaTeX: a\:=\:1\:,\:\:b\:=\:0,\:and\:c\:=\:0.$

The graph of the parent function looks like this:

Parent Quadratic.png

Graphs of a shifted Quadratic Function may look like this:

There are two other forms of quadratic equations:

Vertex Form: $LaTeX: y\:=\:a\left(x-h\right)^2+k$

Intercept (Factored) Form: $LaTeX: y\:=\:a\left(x-p\right)\left(x-q\right)$

Today we are focusing on the Vertex Form:

Vertex Form of a Quadratic and Transformations. Link

You can see from using the sliders that the transformations on the Vertex Form of a Quadratic are identical to those we observed when transforming the Absolute Value function.

Continue exploring this link to note the key (salient) features of a quadratic graph: the vertex, the axis of symmetry, the intercepts (specifically, x - intercepts), and the domain & range.

Key Vocabulary:

Vertex:

the vertex of a parabola will be either the highest (maximum) or lowest (minimum) point on the graph, depending upon whether the graph "opens up" (a > 0) or "opens down" (a < 0)
the vertex of the parent function is (0, 0)
in Vertex Form, we can read the vertex directly from the equation: ( h, k )
notice that the "h" value is always the opposite of what we see in the equation; that is because we are "re-zeroing" the function: if the equation says $LaTeX: \left(x-3\right)^2+7$ , the vertex is at (3, 7) because (x - 3) = 0 when x = 3.

Axis of Symmetry: the axis of symmetry is a vertical line which runs through the center of the parabola, dividing it in "half."

the equation for the axis of symmetry is always x = h (in other words, the x-value of the vertex).
Points on the left and right of the axis of symmetry are reflections across the y -axis.

x - intercept(s): the place(s) where a parabola crosses the x-axis are called intercepts (also called roots or solutions). A parabola may cross the x-axis once, twice, or never. These are a point or points where the y-value is zero, and is/are in the form (p, 0) and (q, 0).

y - intercept: the place where a parabola crosses the y-axis is called the y-intercept. A parabola will only cross the y- axis once. This is a point where the x-value is zero, and is in the form of (0, c).

Domain: the domain of a quadratic function is easy! All real numbers. You can input any number into a quadratic equation.

Range: the range of a quadratic function is almost that easy. Connect the idea of range to two key ideas:

whether the function opens up (a is positive) or down (a is negative)
the +k part of the Vertex Form!
k is either the smallest value, so the range is [k, oo)
or k is the largest value, so the range is (-oo, k]

Tune in to the Math Mad Hatter episode for the day, exploring the connections between quadratic equations in Vertex form and the key features of the graph! Link

Resources:

Introduction to the Vertex Form with the Khan Academy: Link

Another link to a different version of the Desmos example site: Link

A link to the examples in the Math Mad Hatter's Introduction. Make sure you understand the connections between the graphs and the equations: Link

Quadratics Resources Module