Comprehensive Guide to Quadratic Functions and Graphing

A quadratic function is defined by the following standard equation: $f(x) = ax^2 + bx + c$
In this definition, $a$ , $b$ , and $c$ are required to be real numbers.
Requirement for $a$ : The coefficient $a$ cannot be zero ( $a \neq 0$ ). If $a$ were zero, the term $ax^2$ would disappear, leaving the equation as $bx + c$ . This would result in a linear function (a line) rather than a quadratic function.
Terminology for Terms:
- Quadratic Term: The term $ax^2$ is the quadratic term because it is raised to the power of two.
- Linear Term: The term $bx$ is the linear term because it is raised to the power of one.
- Constant Term: The term $c$ is the constant term because it is raised to the power of zero.
The Graph: The specific name for the graph of a quadratic function is a parabola.

Y-intercept: The point where the graph crosses the y-axis is always found at $(0, c)$ . The constant $c$ in the equation directly identifies the y-coordinate of the intercept.
Axis of Symmetry (AOS): This is a vertical line that divides the parabola into two symmetric halves. It is calculated using the formula: $x = -\frac{b}{2a}$
Vertex Calculation: The vertex is the highest or lowest point on the parabola. To find it:
1. Calculate the x-coordinate using the AOS formula ( $x = -\frac{b}{2a}$ ).
2. Plug this x-value back into the function $f(x)$ to find the corresponding y-coordinate.
- The resulting point is $(x, y)$ .
Minimum and Maximum Values: The y-coordinate of the vertex represents the minimum or maximum value of the function.

The direction the parabola opens is determined by the coefficient $a$ :
- Positive $a$ (a > 0): The speaker uses the mnemonic "If you're positive, you're happy. If you're happy, you're smiling." A "smiling" parabola opens upward. When a parabola opens upward, it always has a minimum value.
- Negative $a$ (a < 0): The speaker uses the mnemonic "If you're negative, you're in a bad mood. You're frowning." A "frowning" parabola opens downward. When a parabola opens downward, it always has a maximum value.

Y-intercept: The point where the graph crosses the vertical y-axis.
Axis of Symmetry: A vertical line that passes directly through the vertex, splitting the graph symmetrically.
Vertex: The peak or valley of the graph.
X-intercepts: These are the points where the graph crosses the horizontal x-axis. They are also known as roots, zeros, or solutions. All four terms identify the same property.

Given Function: $f(x) = x^2 + 6x + 8$

Step 1: Identify Constants
- $a = 1$
- $b = 6$
- $c = 8$
Step 2: Find the Y-intercept
- Using $(0, c)$ , the y-intercept is $(0, 8)$ .
Step 3: Calculate the Axis of Symmetry (AOS)
- $x = -\frac{b}{2a}$
- $x = -\frac{6}{2(1)} = -\frac{6}{2} = -3$
- The AOS is the line $x = -3$ .
Step 4: Find the Vertex
- The x-coordinate is $-3$ . Substitue this into the function:
- $f(-3) = (-3)^2 + 6(-3) + 8$
- $f(-3) = 9 - 18 + 8$
- $f(-3) = 17 - 18 = -1$
- The vertex is at $(-3, -1)$ .
Step 5: Create a T-Chart (Table of Values)
- Place the vertex in the center of the chart because the graph is symmetric. Pick points to the left and right of the vertex:
- $x = -3, y = -1$ (Vertex)
- One step away:
  - $f(-2) = (-2)^2 + 6(-2) + 8 = 4 - 12 + 8 = 0$
  - By symmetry, if $x = -2$ yields $y = 0$ , then $x = -4$ also yields $y = 0$ .
- Two steps away:
  - $f(-1) = (-1)^2 + 6(-1) + 8 = 1 - 6 + 8 = 3$
  - By symmetry, if $x = -1$ yields $y = 3$ , then $x = -5$ also yields $y = 3$ .
- Three steps away:
  - $f(0) = (0)^2 + 6(0) + 8 = 8$
  - By symmetry, if $x = 0$ yields $y = 8$ , then $x = -6$ also yields $y = 8$ .

Domain: For all quadratic functions in this chapter, the domain is always $(-\infty, \infty)$ . This is read from left to right (smallest to biggest).
Range: Read from bottom to top.
- In the example $f(x) = x^2 + 6x + 8$ , the lowest point (the bottom) is the vertex y-coordinate: $-1$ .
- The graph goes up toward infinity.
- Because we can actually "achieve" or reach the value $-1$ , we must include it using a bracket: $[-1, \infty)$ .
Intercepts:
- Y-intercept: $(0, 8)$
- X-intercepts: $(-2, 0)$ and $(-4, 0)$
Increasing and Decreasing Intervals: These are determined with respect to the x-axis, pulling values from the x-coordinates.
- Imagine walking on the graph from left to right:
- Decreasing Interval: From the far left until the vertex. In the example, this is $(-\infty, -3)$ .
- Increasing Interval: From the vertex toward the far right. In the example, this is $(-3, \infty)$ .

Dialogue and Instructions: The speaker paused to gather homework materials for those who were not present.
Homework Assignment:
- Problem number 3.
- Problem number 8.
- Directions: Follow the same path and answer the same sequence of questions (intercepts, AOS, vertex, domain, range, etc.) as demonstrated in the example problem.