Lesson 4.1

Quadratic Functions in Vertex and Standard Form

The Parabola is the most famous curve in mathematics. Whether you write it as or changes how easy it is to graph.

Introduction

Prerequisite Connection: In Lesson 2.1, we learned that moves a graph to the point . We are going to apply this directly to quadratics.

Today's Increment: We learn how to convert the "messy" Standard Form () into the "clean" Vertex Form () using Completing the Square.

Why This Matters for Calculus: Optimization problems (finding the maximum profit or minimum cost) often boil down to finding the vertex of a parabola. If you can rewrite the equation, you find the answer instantly.

Explanation of Key Concepts

The Two Forms

Standard Form

Good for finding the y-intercept (0, c).
Bad for visualizing the graph location.
Vertex x-coordinate formula: .

Vertex Form

Good for graphing instantly.
Vertex is at .
Shows horizontal/vertical shifts clearly.

Worked Examples

Level: Intermediate

Example 1: Converting to Vertex Form

Rewrite in vertex form.

Step 1: Focus on x terms

Step 2: Find the "Magic Number"

Take half of b (-6) and square it.

Step 3: Add and Subtract

Answer

Factor the perfect square:

Vertex is at

Level: Conceptual

Example 2: Graphing from Vertex Form

Graph .

h = -1

Shift Left 1

k = 3

Shift Up 3

a = -2

Opens Down, Steep

Level: Advanced

Example 3: Modeling Data

A parabola has a vertex at and passes through . Find its equation.

Start with

Step 1: Plug in Vertex (2, 1)

Step 2: Plug in Point (4, 5)

Step 3: Solve for a

Answer:

Common Pitfalls

Sign Errors with h:
The formula is .
If you see , h is 3 (Shift Right).
If you see , h is -3 (Shift Left).
Always think: "What makes the inside zero?"
Forgetting 'a' when completing the square:
If , you MUST factor out the 2 first: . Don't try to complete the square while the coefficient is there!

Real-World Application

Physics: Trajectories

Any object thrown in a gravitational field (ignoring air resistance) traces a parabola.

If you know the peak height (vertex) and where it lands, Vertex Form is the perfect tool to model the flight path. Engineers use this to calculate where a bridge cable hangs or where a rocket will land.