Lesson 1.5

Standard Form

When the vertex acts shy, we assume Standard Form. We'll use a powerful little formula to find the center of any parabola.

Introduction

Not every quadratic equation reveals its secrets easily. Standard Form is the most common way quadratics appear in algebra, and while it hides the vertex, it gives us powerful tools to find it algebraically.

Past Knowledge

In Vertex Form , looking at the equation tells you the vertex immediately.

Today's Goal

We tackle Standard Form: . The vertex is hidden, so we calculate it.

Future Success

Standard form is the most common format for polynomials and real-world data models.

Key Concepts

Finding the Axis of Symmetry

For , the center line of the parabola is given by:

1
Find the x-coordinateUse the formula to get the x-value of the vertex.
2
Find the y-coordinatePlug that x-value back into the original equation. .
3
The y-interceptThe constant is always the y-intercept . This gives you a free point to graph!

Standard form components visually identified.

Worked Examples

Example 1: Finding the Vertex

Basic

Find the vertex of .

Identify a, b, c

Calculate x

Calculate y

Plug x=2 back in:

Vertex: (2, -1)

Example 2: Downward Parabola

Intermediate

Analyze .

Direction

. Since , it opens DOWN.

Axis of Symmetry

Vertex

Vertex: (2, 3)

Example 3: Graphing Strategy

Strategy

Graph using 3 points.

Vertex

Vertex: (3, -4)

Y-Intercept & Mirror

so y-int is .
Reflect across axis (). The mirror point is .

Common Pitfalls

Forgetting the Negative Sign

The formula is . If is negative (e.g., -4x), double negatives make it positive. becomes +4.

Assuming Vertex is (b, c)

Students sometimes grab random numbers. The vertex x is NOT b. The vertex y is NOT c (that's the y-intercept). You MUST do the calculation.

Real-Life Applications

Maximizing Profit: If a company's profit is modeled by (where x is price), finding the "vertex" tells them exactly what price to charge to make the maximum possible money. The y-coordinate of the vertex IS that maximum profit.