## Learning Objectives:

1. Graph quadratic functions of the form

2. Find the maximum or minimum value of a

quadratic function.

3. Model and solve optimization problems

involving quadratic functions.

## 1. Graphing: The Vertex

The Vertex of a Parabola

Any quadratic function f(x) = ax^2 + bx + c, a ≠ 0, will have vertex

The x- intercepts , if there are any, are found by solving

the quadratic equation

## 1. Graphing: The x-intercepts

The x-intercepts, if there are any, are found by solving

the quadratic equation

**Graphing a Quadratic Function Using Its Properties **

**Step 1:** De termine whether the parabola opens up or
down.

**Step 2: **Determine the vertex and axis of symmetry.

**Step 3:** Determine the y-intercept, f(0).

**Step 4:** Determine the discriminant, b^2 – 4ac.

•If b^2 – 4ac > 0, then the parabola has two x -intercepts, which are
found by solving f(x) = 0.

•If b^2 – 4ac = 0, the vertex is the x-intercept.

•If b^2 – 4ac < 0, there are no x-intercepts.

**Step 5:** Plot the points . Use the axis of symmetry to find an additional
point . Draw the graph of the quadratic function.

## 1. Graphing Using Properties

**Example:**

Graph f(x) = –2x^2 – 8x + 4 using its properties.

## 2. Maximum and Minimum Values

The graph of a quadratic function has a vertex at

The vertex will be the highest point on the graph if a < 0
and will be the **maximum value** of f.

The vertex will be the lowest point on the graph if a > 0
and will be the **minimum value** of f.

## 2. Maximum and Minimum Values

**Example:**

Determine whether the quadratic function

has a maximum or minimum value. Find the value.

## 3. Applications Involving Maximization

**Example:**

The revenue received by a ski resort selling x daily ski lift passes is

given by the function R(x) = – 0.02x^2 + 24x. How many passes

must be sold to maximize the daily revenue?