L4.1 Graphing Quadratic Functions and Finding Maximum/Minimum Values Study Guide

General Purpose: Learning how to graph quadratic functions and how to find and interpret the maximum and minimum values of these functions.
Core Concepts:
* Graphing quadratic functions.
* Identifying, calculating, and interpreting the maximum and minimum values of a quadratic function.

Standard Form: The standard form of a quadratic function is represented as:
* $f(x) = ax^2 + bx + c$ , where $a \neq 0$ .
Components of the Equation:
    * Quadratic term: $ax^2$
    * Linear term: $bx$
    * Constant term: $c$
Critical Characteristic: In a quadratic function, the greatest exponent (the degree) is always 2.
The Parabola: The graph of a quadratic function is a curve called a parabola.

Parabola Definition: To graph a quadratic function, one must graph ordered pairs that satisfy the function and connect them with a smooth curve.
Key Features for Labeling:
    * X-intercepts: The points where the graph intersects the x-axis. These occur when $y = 0$ or $f(x) = 0$ .
    * Y-intercept: The point where the graph intersects the y-axis. This occurs when $x = 0$ . For the standard form $f(x) = ax^2 + bx + c$ , the y-intercept is always $(0, c)$ .
    * Vertex (Turning Point): The highest or lowest point on the graph of the parabola.
    * Axis of Symmetry: A vertical line that passes through the vertex of a parabola and divides the parabola into two congruent halves.

Process:
    1. Choose integer values for $x$ .
    2. Evaluate the function for each chosen value to find the corresponding $f(x)$ (or $y$ ) values.
    3. Graph the resulting coordinate pairs.
    4. Connect the points with a smooth curve.
Example 1: Graph $f(x) = x^2 + 3x - 1$ using a table.
* Calculator Tool: The CASIO fx-991EX CLASSWIZ can be used to generate the ordered pairs using the Table function.
Calculator Instructions (Table Function):
    * Use the MENU SETUP button, navigate to the Table function.
    * Input the function $f(x)$ .
    * Define the start, end, and step values to generate the list of coordinates.

Step 1: Consider the Graph Direction:
* Identify if the parabola opens upward or downward based on the leading coefficient $a$ .
Step 2: Find the Axis of Symmetry:
* Use the formula: $x = -\frac{b}{2a}$ .
Step 3: Find the Vertex:
    * The vertex lies on the axis of symmetry, so its x-coordinate is $-\frac{b}{2a}$ .
    * Find the y-coordinate by substituting the x-coordinate back into the original function.
    * Example: For an axis of symmetry at $x = -3$ , substitute $-3$ into the equation to find the y-value (e.g., $y = -11$ ), resulting in a vertex of $(-3, -11)$ .
Step 4: Find the Y-intercept:
* Substitute $x = 0$ into the equation.
* Example: For $f(x) = x^2 + 6x - 2$ , if $x = 0$ , then $y = -2$ . The point is $(0, -2)$ .
Step 5: Find the Reflected Point:
* Use the axis of symmetry to find the reflection of the y-intercept.
* Example: If the y-intercept is $(0, -2)$ and the axis of symmetry is at $x = -3$ , the reflected point is $(-6, -2)$ .
Step 6: Connect the Points:
* Draw a smooth curve through the vertex, the y-intercept, and the reflected point.

Maximum and Minimum Rule:
* If a > 0: The graph opens upward and has a minimum value.
* If a < 0: The graph opens downward and has a maximum value.
Defining the Value: The actual maximum or minimum value of the function is the y-coordinate of the vertex.
Example 4: Consider $f(x) = -x^2 + 2x + 3$ .
    * $a = -1, b = 2, c = 3$ .
    * Since a < 0, the graph opens downward; the function has a maximum value.
    * The vertex is found at $x = -\frac{2}{2(-1)} = 1$ .
    * Evaluating $f(1) = -(1)^2 + 2(1) + 3 = 4$ .
    * The maximum value is $4$ .

Domain: The set of all possible input values ( $x$ -values). For any quadratic function, the domain is all real numbers.
Range: The set of all possible output values ( $y$ -values).
* If the graph opens upward (has a minimum): Range is $f(x) \geq \text{y-value of the vertex}$ .
* If the graph opens downward (has a maximum): Range is $f(x) \leq \text{y-value of the vertex}$ .
Example 5: For $f(x) = -x^2 + 2x + 3$ (where the maximum is $4$ ):
* Domain: All real numbers.
* Range: $f(x) \leq 4$ .

Analyze Coefficients: $a = 2, b = -4, c = 5$ .
A) Direction: Opens upward because $a = 2$ (positive).
B) Axis of Symmetry: $x = -\frac{-4}{2(2)} = \frac{4}{4} = 1$ . The line is $x = 1$ .
C) Vertex: Substitute $x = 1$ into the function: $f(1) = 2(1)^2 - 4(1) + 5 = 2 - 4 + 5 = 3$ . Vertex is $(1, 3)$ .
D) Y-intercept: Since $c = 5$ , the y-intercept is $(0, 5)$ .
E) Graphing:
    * Plot $(1, 3)$ .
    * Plot $(0, 5)$ .
    * Plot the symmetric point. Since $(0, 5)$ is $1$ unit left of the axis $x = 1$ , the symmetric point is $1$ unit right: $(2, 5)$ .

Scenario: A rock is thrown upward from a $112\,foot$ high cliff at a speed of $96\,feet/second$ .
Model Equation: $H(t) = -16t^2 + 96t + 112$ .
Question: When does the rock hit the ocean?
Solution:
    * When the rock hits the ocean, its height $H(t) = 0$ .
    * Equation: $0 = -16t^2 + 96t + 112$ .
    * Solve for zeros using a calculator.
    * Results: $t = 7$ and $t = -1$ .
    * Conclusion: Ignore the negative value. The rock hits the ocean after $7\,seconds$ .

Search and Synergize (Introductory Activity):
    * 1) What is the quadratic function?
    * 2) Write the standard Form of a quadratic function?
    * 3) What does the graph of a quadratic function look like?
    * 4) How to graph a quadratic function using a table?
Search and Synergize (Advanced Activity):
* 1) What are the maximum and minimum values of a quadratic function?
* 2) What are the domain and range of a quadratic function?
EST Practice 1: Which graph is the graph of $f(x) = 2x^2 + 3x + 2$ ?
EST Practice 2: Consider $f(x) = 3 - 6x + x^2$ . Find the y-intercept, axis of symmetry, and x-coordinate of the vertex.
* Result: y-intercept = $3$ ; Axis of symmetry: $x = 3$ ; x-coordinate = $3$ (Option C).
EST Practice 3: Consider $f(x) = 3 - 6x + x^2$ . Make a table of values that includes the vertex.
EST Practice 4: Use information from parts A and B to graph the function $f(x) = 3 - 6x + x^2$ .
EST Practice 5: Consider $f(x) = x^2 + 4x - 1$ . Determine if it has a max or min.
* Result: Minimum at $x = -2$ (Option B).
EST Practice 6: What is the maximum or minimum value of $f(x) = x^2 + 4x - 1$ ?
* Calculation: $x = -4/2 = -2$ . $f(-2) = (-2)^2 + 4(-2) - 1 = 4 - 8 - 1 = -5$ .
* Result: $-5$ (Option A).
EST Practice 7: What are the domain and range of $f(x) = x^2 + 4x - 1$ ?
* Result: $D = {\text{all real numbers}}; R = {f(x) | f(x) \geq -5}$ (Option A).