Lesson 1.6

Intercept Form

Sometimes the most important thing isn't where the peak is, but where we hit the ground. Intercept form gives us the roots instantly.

Introduction

Sometimes, where we start and where we finish is more important than how high we go. Intercept Form focuses on the 'roots' or zeros of the function, making it ideal for solving problems about landing points and break-even analysis.

Past Knowledge

You can graph parabolas using the Vertex (Vertex Form) or the formula (Standard Form).

Today's Goal

We explore Intercept Form: . This form instantly reveals where the graph crosses the x-axis.

Future Success

This form connects directly to factoring quadratics, finding "zeros" or "roots," and solving .

Key Concepts

The Intercept Form Equation

X
The Intercepts (p, 0) and (q, 0)Solve and . The graph crosses the x-axis at and .
A
Axis of Symmetry (Midpoint)Parabolas are symmetrical. The center line is exactly halfway between the intercepts.
x = (p + q) / 2
V
Finding the VertexOnce you have the axis of symmetry x-value, plug it back into the equation to find y.

Intercepts at 1 and 5. Axis at 3.

Worked Examples

Example 1: Graphing from Intercepts

Basic

Graph .

Identify Intercepts

Set factors to zero: and .
Points: (-2, 0) and (4, 0).

Find Axis of Symmetry

Find Vertex Y

Plug x=1 back in:

Vertex: (1, 9)

Example 2: Just x?

Concept

Analyze .

Intercepts

Factors are and .
So and .

Midpoint

Vertex

Vertex: (3, -18)

Example 3: Working Backwards

Analysis

Find the equation of a parabola with intercepts at and that passes through .

Set up Intercept Form

Solve for 'a'

Plug in :

Equation:

Common Pitfalls

Sign Confusion

has intercepts at -2 and -5, NOT 2 and 5. Always set the factor equal to zero: .

Forgetting the 'a'

When finding coordinates, always check for the negative sign or number in front. opens down!

Real-Life Applications

Bridge Construction: When engineers design a suspension bridge, they know the span. If supports are at 0 meters and 100 meters, the "intercepts" are 0 and 100. The equation models the main cable perfectly. The 'a' value is determined by how low the cable sags in the middle (vertex).