# Introduction to Quadratic Functions

## A. Definition

is **
a quadratic function if
where a≠0**

The graph is a **parabola**

## B. Parabolas

1. Parabolas

2. v is the **vertex** of the parabola

## C. Formula for the Vertex

1. Vertex Formula: |

2. See **MTH103: College Algebra** for the
justification of this formula

## D. Graphing Quadratic Functions (Parabolas)

1. Locate the vertex: use the vertex formula

2. Find the x and y intercepts.

3. Plot the points and connect in a smooth curve, recognizing whether it opens
up or down.

## E. Examples of Graphing

**Example 1:** Graphwhere

Solution

1. Locate vertex:

Vertex Formula:

Thus the vertex is (-3,-3)

2. Intercepts

y-intercept:

x-intercept:

3. Note the graph opens upward:

**Example 2:** Graphwhere

Solution

1. Locate vertex:

Vertex Formula:

Thus the vertex is (1,-5)

2. Intercepts

y-intercept:

x-intercept:

no real solutions!

Thus, no x-intercept

3. Note the graph opens downward:

## F. Projectile Motion

A thrown object follows a parabolic path given by

initial upward speed

initial height thrown from

Units of height: feet

The vertex is the peak of the path.

**Features**

1. The t-coordinate (“ x-coordinate”) of the vertex tells
you

**when** the projectile reaches its maximum height

2. The “y-coordinate” of the vertex tells you what the maximum height is.

3. The projectile hits the ground when

## G. Projectile Motion Example

A ball is thrown upward with a speed of 16ft/s from a
building 32 feet high.

What is the maximum height of the ball? When does it reach the ground?

Solution

Use the projectile motion model:

Now

Also,

Thus the vertex is (1/2,36)

Now the maximum height is the y-coordinate of the vertex:
**36 feet**

The ball reaches the ground when

Since t=1 is not physical, we must have t=2 .

**Thus the ball reaches the ground 2 seconds after being
thrown.**