Graphing Quadratics

🎢 Graphing Quadratics: The Magic of U-Shaped Curves

Imagine throwing a ball in the air. It goes up, slows down, and comes back down. That path? It’s a parabola—and today, you’ll learn to draw one yourself!

🌟 The Big Picture

A quadratic function is like a recipe that makes a special U-shaped curve called a parabola.

The recipe looks like this:

y = ax² + bx + c

Think of it as a magic formula. You put in any number for x, and out comes y. Plot enough points, and boom—you get a beautiful curve!

🎯 What is a Parabola?

Imagine a fountain. Water shoots up from the center, curves gracefully at the top, then falls back down on both sides. That curved path is a parabola!

Quick Facts:

• Opens UP when a is positive (like a smile 😊)
• Opens DOWN when a is negative (like a frown ☹️)
• Every parabola has a mirror line running through its center

Example:

• y = x² → Opens UP (a = 1, positive)
• y = -x² → Opens DOWN (a = -1, negative)

📍 The Vertex: The Star of the Show

The vertex is the turning point—where the parabola changes direction.

Think of it this way:

If you’re climbing a hill, the vertex is the very top (or very bottom if you’re in a valley). It’s the highest or lowest point on the curve.

Finding the Vertex

Step 1: Find the x-coordinate using this formula:

x = -b ÷ (2a)

Step 2: Plug that x back into the equation to get y.

🎯 Example: y = x² - 4x + 3

Given: a = 1, b = -4, c = 3

Find x:

x = -(-4) ÷ (2 × 1)
x = 4 ÷ 2
x = 2

Find y: Plug x = 2 back in:

y = (2)² - 4(2) + 3
y = 4 - 8 + 3
y = -1

Vertex = (2, -1) 🎉

This point is the lowest spot on our parabola!

⚖️ The Axis of Symmetry: The Mirror Line

Every parabola has a secret mirror line running straight through its vertex. This is the axis of symmetry.

The Magic:

Whatever happens on the LEFT of this line is perfectly reflected on the RIGHT!

The Formula:

x = -b ÷ (2a)

Yes! It’s the same formula as the x-coordinate of the vertex!

🎯 Example: y = 2x² + 8x + 5

Given: a = 2, b = 8

x = -8 ÷ (2 × 2)
x = -8 ÷ 4
x = -2

Axis of Symmetry: x = -2

Draw a vertical line at x = -2, and the parabola folds perfectly onto itself!

📈 Maximum and Minimum Values

When the Parabola Opens UP (a > 0):

• The vertex is at the BOTTOM
• This is the MINIMUM value
• The parabola can go up forever, but never below the vertex

When the Parabola Opens DOWN (a < 0):

• The vertex is at the TOP
• This is the MAXIMUM value
• The parabola can go down forever, but never above the vertex

graph TD
    A["Look at 'a' in y = ax² + bx + c"] --> B{Is a positive?}
    B -->|Yes| C["Opens UP ⬆️"]
    B -->|No| D["Opens DOWN ⬇️"]
    C --> E["Vertex is MINIMUM"]
    D --> F["Vertex is MAXIMUM"]

🎯 Example 1: y = x² - 6x + 8

• a = 1 (positive) → Opens UP
• Vertex x = -(-6)/(2×1) = 3
• Vertex y = 9 - 18 + 8 = -1
• Minimum value = -1 at x = 3

🎯 Example 2: y = -x² + 4x - 3

• a = -1 (negative) → Opens DOWN
• Vertex x = -4/(2×-1) = 2
• Vertex y = -4 + 8 - 3 = 1
• Maximum value = 1 at x = 2

🎨 How to Graph a Quadratic Function

Follow these steps to draw any parabola:

Step 1: Find the Vertex

Use x = -b/(2a), then find y.

Step 2: Draw the Axis of Symmetry

Draw a dotted vertical line through the vertex.

Step 3: Find the Y-Intercept

Set x = 0 and solve for y. This is point (0, c).

Step 4: Find More Points

Pick x values on one side of the axis, calculate y, then mirror them!

Step 5: Connect with a Smooth Curve

Draw a nice U-shape through all your points.

🚀 Graphing Example: y = x² - 2x - 3

Step 1: Vertex

• x = -(-2)/(2×1) = 1
• y = 1 - 2 - 3 = -4
• Vertex: (1, -4)

Step 2: Axis of Symmetry

• x = 1

Step 3: Y-Intercept

• When x = 0: y = 0 - 0 - 3 = -3
• Y-intercept: (0, -3)

Step 4: More Points

Step 5: Plot & Connect!

Notice: (0, -3) and (2, -3) are mirror images across x = 1!

🧠 The Big Ideas to Remember

🌈 Real-World Parabolas

Parabolas are everywhere!

• 🏀 Basketball shots follow parabolic paths
• 🌉 Bridge cables hang in parabolas
• 📡 Satellite dishes are shaped like parabolas
• ⛲ Fountains spray water in parabolic arcs
• 🎆 Fireworks explode in parabolic patterns

✨ Your Superpower Unlocked!

You now know how to:

1. ✅ Find the vertex of any parabola
2. ✅ Draw the axis of symmetry
3. ✅ Identify maximum and minimum values
4. ✅ Graph any quadratic function step by step

The parabola isn’t just a shape—it’s a pattern that appears throughout nature and engineering. And now, you can see it, understand it, and draw it yourself!

You’re ready to make those curves dance! 🎢✨