# Polynomial and Rational Functions

## Quadratic Functions


### Learning Objectives

1. Graph quadratic functions using properties. (IA 9.6.4)


### Objective 1: Graph quadratic functions using properties. (IA 9.6.4)

A quadratic function is a function that can be written in the general form , where a, b, and c are real numbers and a≠0. The graph of quadratic function is called a parabola. Parabolas are symmetric around a line (also called an axis) and have the highest (maximum) or the lowest (minimum) point that is called a vertex.


### Making a Table

We can graph quadratic function  by making a table and plotting points.


### Practice Makes Perfect

Graph quadratic function by making a table and plotting points.
Graphing of quadratic functions is much easier when we know the vertex and the axis of symmetry. The vertex of the graph of the quadratic function  in the form  .
The line or axis of symmetry of the parabola is the vertical line  .


### Practice Makes Perfect

Graphing quadratic functions using a vertex.
Curved antennas, such as the ones shown in , are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

### Recognizing Characteristics of Parabolas 

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in . 
The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of 
  at which 
  

### Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

The general form of a quadratic function presents the function in the form
where 
  and 
  are real numbers and 
  If 
  the parabola opens upward. If 
  the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.
The axis of symmetry is defined by 
  If we use the quadratic formula, 
  to solve 
  for the 
  intercepts, or zeros, we find the value of 
  halfway between them is always 
  the equation for the axis of symmetry.
 represents the graph of the quadratic function written in general form as 
  In this form, 
  and 
  Because 
  the parabola opens upward. The axis of symmetry is 
  This also makes sense because we can see from the graph that the vertical line 
  divides the graph in half. The vertex always occurs along the axis of symmetry.  For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, 
  The 
  intercepts, those points where the parabola crosses the 
  axis, occur at 
  and 
  
The standard form of a quadratic function presents the function in the form
where 
  is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
As with the general form, if 
  the parabola opens upward and the vertex is a minimum. If 
  the parabola opens downward, and the vertex is a maximum.  represents the graph of the quadratic function written in standard form as 
  Since 
  in this example, 
  In this form, 
  and 
  Because 
  the parabola opens downward. The vertex is at 
  
The standard form is useful for determining how the graph is transformed from the graph of 
   is the graph of this basic function.
If 
  the graph shifts upward, whereas if 
  the graph shifts downward. In , 
  so the graph is shifted 4 units upward. If 
  the graph shifts toward the right and if 
  the graph shifts to the left. In , 
  so the graph is shifted 2 units to the left. The magnitude of 
  indicates the stretch of the graph. If 
  the point associated with a particular 
  value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if 
  the point associated with a particular 
  value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In , 
  so the graph becomes narrower.
The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.
For the linear terms to be equal, the coefficients must be equal.
This is the axis of symmetry we defined earlier. Setting the constant terms equal:
In practice, though, it is usually easier to remember that k is the output value of the function when the input is 
  so 
  

### Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.

### Determining the Maximum and Minimum Values of Quadratic Functions

The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola. We can see the maximum and minimum values in .
There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

### Finding the x- and y-Intercepts of a Quadratic Function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the 
  intercept of a quadratic by evaluating the function at an input of zero, and we find the 
  intercepts at locations where the output is zero. Notice in  that the number of 
  intercepts can vary depending upon the location of the graph.

### Rewriting Quadratics in Standard Form

In , the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

### Key Equations


### Key Concepts

1. A polynomial function of degree two is called a quadratic function.
2. The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
3. The axis of symmetry is the vertical line passing through the vertex. The zeros, or 
  intercepts, are the points at which the parabola crosses the 
  axis. The 
  intercept is the point at which the parabola crosses the 
  axis. See , , and .
4. Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See .
5. The vertex can be found from an equation representing a quadratic function. See .
6. The domain of a quadratic function is all real numbers. The range varies with the function. See .
7. A quadratic function’s minimum or maximum value is given by the 
  value of the vertex.
8. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See  and .
9. The vertex and the intercepts can be identified and interpreted to solve real-world problems. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
For the following exercises, determine the domain and range of the quadratic function.
For the following exercises, use the vertex 
  and a point on the graph 
  to find the general form of the equation of the quadratic function.

### Graphical

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.
For the following exercises, write the equation for the graphed quadratic function.

### Numeric

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

### Technology

For the following exercises, use a calculator to find the answer.

### Extensions

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.
For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

### Real-World Applications
