Vertex Form and Transformations
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn vertex form of a quadratic function and how changes in the equation (a, h, k) affect the graph through translations, stretches, and reflections.
The Concept
Section titled “The Concept”Vertex form of a quadratic function is:
Where:
- The vertex is at the point (h, k).
- a tells the direction and width:
- a > 0 opens upward
- a < 0 opens downward
- |a| > 1 makes the parabola narrower (steeper)
- |a| < 1 makes the parabola wider (flatter)
Transformations from the parent function y = x²:
- Horizontal shift: (x − h) shifts the graph right by h units (left if h is negative)
- Vertical shift: +k shifts the graph up by k units (down if k is negative)
- Vertical stretch/compression: |a| affects how tall or wide the parabola is
- Reflection: a negative a reflects the parabola over the x-axis
Worked Example
Section titled “Worked Example”1. Convert standard form to vertex form
Write f(x) = x² − 6x + 5 in vertex form by completing the square.
Completing the square is a technique for rewriting a quadratic expression so that the x terms form a perfect square trinomial. The idea is to take the coefficient of x, divide it by 2, square the result, and add/subtract that number to create a perfect square. Here’s how it works step by step:
Start with the expression x² − 6x + 5.
Step 1: Focus on the x² − 6x part. Take the coefficient of x (which is −6), divide by 2 to get −3, then square it to get 9.
Step 2: Add and subtract 9 inside the expression. Adding 9 creates the perfect square, and subtracting 9 keeps the value the same:
Step 3: The first group is now a perfect square trinomial. It factors as (x − 3)²:
Vertex form: f(x) = (x − 3)² − 4. The vertex is (3, −4).
2. Describe the transformations
Describe the transformations for g(x) = −2(x + 1)² + 5.
The dashed gray curve is the parent function y = x², with its vertex at the origin. The solid orange curve is g(x) = −2(x + 1)² + 5. The green arrow shows how the vertex moved from (0, 0) to (−1, 5). Here’s what each part of the equation does:
- Reflected over the x-axis (the negative sign flips the U upside down)
- Vertically stretched by a factor of 2 (the parabola is narrower/steeper than y = x²)
- Shifted left 1 unit (the (x + 1) means h = −1)
- Shifted up 5 units (the +5 means k = 5)
The vertex is (−1, 5), which is the maximum since the parabola opens downward.
Real-World Application
Section titled “Real-World Application”Vertex form is very useful in real life:
- Projectile motion: height of a ball or rocket. The vertex gives maximum height.
- Business: profit function. Vertex gives maximum profit and the number of units to produce.
- Architecture: parabolic arches and bridges. Vertex form helps calculate height and width.
- Physics: path of a thrown object or satellite dish shape.
Example: a ball is thrown with height h(t) = −16t² + 48t + 6. Converting to vertex form tells you the maximum height and when it occurs.