# 10.11: Linear, Exponential, and Quadratic Models

**At Grade**Created by: CK-12

**Practice**Linear, Exponential, and Quadratic Models

### Linear, Exponential, and Quadratic Models

In this course we’ve learned about three types of functions, linear, quadratic and exponential.

- Linear functions take the form
y=mx+b . - Quadratic functions take the form
y=ax2+bx+c . - Exponential functions take the form
y=a⋅bx .

In real-world applications, the function that describes some physical situation is not given; it has to be found before the problem can be solved. For example, scientific data such as observations of planetary motion are often collected as a set of measurements given in a table. Part of the scientist’s job is to figure out which function best fits the data. In this section, you’ll learn some methods that are used to identify which function describes the relationship between the variables in a problem.

**Identify Functions Using Differences or Ratios**

One method for identifying functions is to look at the difference or the ratio of different values of the dependent variable. For example, **if the difference between values of the dependent variable is the same each time we change the independent variable by the same amount, then the function is** *linear***.**

#### Identifying Functions

Determine if the function represented by the following table of values is linear.

–4 | |

–1 | –1 |

0 | 2 |

1 | 5 |

2 | 8 |

If we take the difference between consecutive

Since the difference is always the same, **the function is linear.**

When we look at the difference of the

For example, examine the values in this table:

0 | 5 |

1 | 10 |

3 | 20 |

4 | 25 |

6 | 35 |

At first glance this function might not look linear, because the difference in the *same* amount, the

Another way to think of this is in mathematical notation. We can say that a function is linear if

Differences can also be used to identify quadratic functions. **For a quadratic function, when we increase the x−values by the same amount, the difference between y−values will not be the same. However, the difference of the**

*differences***of the**y− values will be the same.

Here are some examples of quadratic relationships represented by tables of values:

In this quadratic function,

In this quadratic function,

To identify exponential functions, we use ratios instead of differences. **If the ratio between values of the dependent variable is the same each time we change the independent variable by the same amount, then the function is** *exponential.*

#### Determining if a Function is Exponential

Determine if the function represented by each table of values is exponential.

a)

If we take the ratio of consecutive **the function is exponential.**

b)

If we take the ratio of consecutive **the function is exponential.**

**Writing Equations for Functions**

Once we identify which type of function fits the given values, we can write an equation for the function by starting with the general form for that type of function.

1. Determine what type of function represents the values in the following table.

0 | 5 |

1 | 1 |

2 | -3 |

3 | -7 |

4 | -11 |

Let’s first check the difference of consecutive values of

If we take the difference between consecutive **the function is linear.**

To find the equation for the function, we start with the general form of a linear function:

2. Determine what type of function represents the values in the following table.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 0 |

1 | 5 |

2 | 20 |

3 | 45 |

4 | 80 |

5 | 125 |

6 | 180 |

Here, the difference between consecutive \begin{align*}y-\end{align*}values isn’t constant, so the function is not linear. Let’s look at those differences more closely.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} | |
---|---|---|

0 | 0 | |

1 | 5 | \begin{align*}5-0=5\end{align*} |

2 | 20 | \begin{align*}20-5=15\end{align*} |

3 | 45 | \begin{align*}45-20=25\end{align*} |

4 | 80 | \begin{align*}80-45=35\end{align*} |

5 | 125 | \begin{align*}125-80=45\end{align*} |

6 | 180 | \begin{align*}180-125=55\end{align*} |

When the \begin{align*}x-\end{align*}value increases by one, the difference between the \begin{align*}y-\end{align*}values increases by 10 every time. Since the difference of the differences is constant, the function describing this set of values is **quadratic.**

To find the equation for the function that represents these values, we start with the general form of a quadratic function: \begin{align*}y=ax^2+bx+c\end{align*}.

We need to use the values in the table to find the values of the constants: \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}.

The value of \begin{align*}c\end{align*} represents the value of the function when \begin{align*}x = 0\end{align*}, so \begin{align*}c = 0\end{align*}.

\begin{align*}\text{Plug in the point} \ (1, 5): && 5 &= a+b\\ \text{Plug in the point} \ (2, 20): && 20 &= 4a+2b \Rightarrow 10=2a+b\\ \text{To find} \ a \ \text{and} \ b, \text{we solve the system of equations:} && 5 &= a+b\\ && 10 &= 2a+b\\ \text{Solve the first equation for} \ b: && 5 &= a+b \Rightarrow b=5-a\\ \text{Plug the first equation into the second:} && 10 &= 2a+5-a\\ \text{Solve for} \ a \ \text{and} \ b && a &= 5 \ \text{and} \ b=0\end{align*}

Therefore the equation of the quadratic function is \begin{align*}y=5x^2\end{align*}.

### Example

#### Example 1

Determine what type of function represents the values in the following table.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 400 |

1 | 500 |

2 | 25 |

3 | 6.25 |

4 | 1.5625 |

The differences between consecutive \begin{align*}y-\end{align*}values aren’t the same, and the differences between those differences aren’t the same either. So let’s check the ratios instead.

Each time the \begin{align*}x-\end{align*}value increases by one, the \begin{align*}y-\end{align*}value is multiplied by \begin{align*}\frac{1}{4}\end{align*}. Since the ratio is always the same, **the function is exponential.**

To find the equation for the function that represents these values, we start with the general form of an exponential function, \begin{align*}y=a \cdot b^x\end{align*}.

Here \begin{align*}b\end{align*} is the ratio between the values of \begin{align*}y\end{align*} each time \begin{align*}x\end{align*} is increased by one. The constant \begin{align*}a\end{align*} is the value of the function when \begin{align*}x = 0\end{align*}. Therefore, the function is \begin{align*}y=400 \left( \frac{1}{4} \right)^x\end{align*}.

### Review

Determine whether the data in the following tables can be represented by a linear function.

1.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-4\end{align*} | 10 |

-3 | 7 |

-2 | 4 |

-1 | 1 |

0 | -2 |

1 | -5 |

2.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-2\end{align*} | 4 |

-1 | 3 |

0 | 2 |

1 | 3 |

2 | 6 |

3 | 11 |

3.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 50 |

1 | 75 |

2 | 100 |

3 | 125 |

4 | 150 |

5 | 175 |

Determine whether the data in the following tables can be represented by a quadratic function.

4.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-10\end{align*} | 10 |

-5 | 2.5 |

0 | 0 |

5 | 2.5 |

10 | 10 |

15 | 22.5 |

5.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

1 | 4 |

2 | 6 |

3 | 6 |

4 | 4 |

5 | 0 |

6 | -6 |

6.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | -27 |

-2 | -8 |

-1 | -1 |

0 | 0 |

1 | 1 |

2 | 8 |

3 | 27 |

Determine whether the data in the following tables can be represented by an exponential function.

7.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 200 |

1 | 300 |

2 | 1800 |

3 | 8300 |

4 | 25800 |

5 | 62700 |

8.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 120 |

1 | 180 |

2 | 270 |

3 | 405 |

4 | 607.5 |

5 | 911.25 |

9.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 4000 |

1 | 2400 |

2 | 1440 |

3 | 864 |

4 | 518.4 |

5 | 311.04 |

Determine what type of function represents the values in the following tables and find the equation of each function.

10.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

0 | 400 |

1 | 500 |

2 | 625 |

3 | 781.25 |

4 | 976.5625 |

11.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-9\end{align*} | -3 |

-7 | -2 |

-5 | -1 |

-3 | 0 |

-1 | 1 |

1 | 2 |

12.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

\begin{align*}-3\end{align*} | 14 |

-2 | 4 |

-1 | -2 |

0 | -4 |

1 | -2 |

2 | 4 |

3 | 14 |

### Review (Answers)

To view the Review answers, open this PDF file and look for section 10.11.

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### Image Attributions

Here you'll learn how to identify a function's type by examining the difference or the ratio of different values of the dependent variable.

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