Materials Needed

• Graph paper and a ruler (or access to Desmos Graphing Calculator)
• Three different colored pens or pencils
• A small ball (ping pong ball, tennis ball, or crumpled paper)
• Smartphone or tablet with a camera
• Worksheet or digital document for data entry

Learning Objectives

By the end of this lesson, you will be able to:

• Identify the Standard Form of a quadratic function and the roles of its coefficients.
• Calculate and graph the Vertex, Axis of Symmetry, and Zeros of a parabola.
• Analyze how changing the leading coefficient affects the "width" and direction of the graph.
• Model a real-world projectile path using a quadratic equation.

1. Introduction: The Hook (10 Minutes)

The Scenario: Have you ever wondered why basketball players shoot in an arc rather than a straight line? Or why the "Golden Arches" of McDonald's look the way they do? These aren't just random curves; they are parabolas, the visual representation of quadratic functions.

The Mini-Experiment: Place your phone on a table and record a video in slow motion. Toss a small ball in the air to a target (like a trash can or a chair) about 5 feet away. Watch the playback. Think-Pair-Share (or Self-Reflection): What happened to the speed of the ball at the very top of the arc? If you drew a line down the middle of the path, would the left side match the right?

2. Instruction: "I Do" - Decoding the Quadratic (15 Minutes)

Every parabola comes from an equation in Standard Form: y = ax² + bx + c

• The "a" value: This is the "shape-shifter." If a is positive, the graph smiles (opens up). If a is negative, it frowns (opens down). If a is a big number, the graph is skinny. If a is a tiny fraction, the graph is wide.
• The "c" value: This is the y-intercept. It's where the ball starts in your hand before you throw it.
• The Vertex: This is the "turning point" (the peak of your throw). We find its x-coordinate using the formula: x = -b / 2a.

Example Modeling: Let's look at y = -x² + 4x + 5.
1. a = -1 (opens down), b = 4, c = 5.
2. Vertex x-coord: -4 / (2 * -1) = 2.
3. Plug 2 back in to find y: -(2)² + 4(2) + 5 = 9. Vertex is (2, 9).

3. Guided Practice: "We Do" - Building the Curve (20 Minutes)

Let's graph the function f(x) = x² - 2x - 3 together.

1. Identify a, b, and c: a=1, b=-2, c=-3.
2. Find the Axis of Symmetry: Use x = -b/2a. (Result: x = 1). Draw a dashed vertical line here with your first colored pencil.
3. Find the Vertex: Plug x=1 into the equation. (Result: (1, -4)). Plot this point.
4. Find the Y-Intercept: It's just the 'c' value. (Result: (0, -3)). Plot this.
5. Find the Roots (Zeros): Where does the graph hit the floor? Set y to 0 and factor: (x-3)(x+1) = 0. (Results: x = 3 and x = -1). Plot these.
6. Connect the dots: Draw a smooth, U-shaped curve through these points.

4. Independent Application: "You Do" - The Skate Park Challenge (25 Minutes)

The Task: You are designing a "Half-Pipe" ramp for a local skate park. To be safe, the ramp must follow a specific quadratic path.

Your Constraints:

• The bottom of the ramp (the vertex) must be at (0, 0).
• The ramp must pass through the points (-4, 8) and (4, 8) so skaters can enter from a platform 8 feet high.

Action Steps:

1. Determine the equation in the form y = ax² that fits these points. (Hint: Plug in 4 for x and 8 for y to solve for a).
2. Create a table of values for x = -4, -2, 0, 2, 4.
3. Graph your ramp accurately on graph paper.
4. Analysis: If you wanted to make the ramp "steeper" for professional skaters, would you increase or decrease the a value? Write a one-sentence justification.

5. Conclusion & Assessment (10 Minutes)

Recap: We learned that quadratics aren't just numbers—they describe gravity and design. We found that the vertex is the most important point on the graph and that 'a' controls the "vibe" of the curve.

Success Criteria Check (Summative Assessment):

• Can you find the vertex of y = x² - 6x + 8?
• Does the graph of y = -2x² + 5 open up or down?
• If a parabola is perfectly symmetrical, and one root is at x=2 and the vertex is at x=4, where is the other root?

Differentiation & Extensions

• For Visual Learners: Use Desmos to animate the "a" coefficient. Create a slider for 'a' and watch the parabola "dance" as the number changes.
• For Advanced Learners: Try converting an equation from Standard Form to Vertex Form y = a(x-h)² + k by completing the square. How does this make graphing faster?
• For Scaffolding: Use a pre-printed table of values where the x-coordinates are already chosen to help focus on the calculation of y-values.