# Lesson 3

Equations for Functions

Let’s find outputs from equations.

### 3.1: A Square’s Area

Fill in the table of input-output pairs for the given rule. Write an algebraic expression for the rule in the box in the diagram.

input | output |
---|---|

8 | |

2.2 | |

\(12\frac14\) | |

\(s\) |

### 3.2: Diagrams, Equations, and Descriptions

Record your answers to these questions in the table provided.

- Match each of these descriptions with a diagram:
- the circumference, \(C\), of a circle with
**radius**, \(r\) - the distance in miles, \(d\), that you would travel in \(t\) hours if you drive at 60 miles per hour
- the output when you triple the input and subtract 4
- the volume of a cube, \(v\) given its edge length, \(s\)

- the circumference, \(C\), of a circle with
- Write an equation for each description that expresses the output as a function of the input.
- Find the output when the input is 5 for each equation.
- Name the
**independent**and**dependent variables**of each equation.

description | a | b | c | d |
---|---|---|---|---|

diagram | ||||

equation | ||||

input = 5 output = ? |
||||

independent variable |
||||

dependent variable |

Choose a 3-digit number as an input.

Apply the following rule to it, one step at a time:

- Multiply your number by 7.
- Add one to the result.
- Multiply the result by 11.
- Subtract 5 from the result.
- Multiply the result by 13
- Subtract 78 from the result to get the output.

Can you describe a simpler way to describe this rule? Why does this work?

### 3.3: Dimes and Quarters

Jada had some dimes and quarters that had a total value of $12.50. The relationship between the number of dimes, \(d\), and the number of quarters, \(q\), can be expressed by the equation \(0.1d + 0.25q = 12.5\).

- If Jada has 4 quarters, how many dimes does she have?
- If Jada has 10 quarters, how many dimes does she have?
- Is the number of dimes a function of the number of quarters? If yes, write a rule (that starts with \(d = \)...) that you can use to determine the output, \(d\), from a given input, \(q\). If no, explain why not.
- If Jada has 25 dimes, how many quarters does she have?
- If Jada has 30 dimes, how many quarters does she have?
- Is the number of quarters a function of the number of dimes? If yes, write a rule (that starts with \(q=\)...) that you can use to determine the output, \(q\), from a given input, \(d\). If no, explain why not.

### Summary

We can sometimes represent functions with equations. For example, the area, \(A\), of a circle is a function of the radius, \(r\), and we can express this with an equation: \(\displaystyle A=\pi r^2\)

We can also draw a diagram to represent this function:

In this case, we think of the radius, \(r\), as the input, and the area of the circle, \(A\), as the output. For example, if the input is a radius of 10 cm, then the output is an area of \(100\pi\) cm^{2}, or about 314 square cm. Because this is a function, we can find the area, \(A\), for any given radius, \(r\).

Since it is the input, we say that \(r\) is the **independent variable** and, as the output, \(A\) is the **dependent variable**.

Sometimes when we have an equation we get to choose which variable is the independent variable. For example, if we know that

\(\displaystyle 10A-4B=120\)

then we can think of \(A\) as a function of \(B\) and write

\(\displaystyle A=0.4B+12\)

or we can think of \(B\) as a function of \(A\) and write

\(\displaystyle B=2.5A-30\)

### Glossary Entries

**dependent variable**A dependent variable represents the output of a function.

For example, suppose we need to buy 20 pieces of fruit and decide to buy apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of bananas is the dependent variable because it depends on the number of apples.

**independent variable**An independent variable represents the input of a function.

For example, suppose we need to buy 20 pieces of fruit and decide to buy some apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of apples is the independent variable because we can choose any number for it.

**radius**A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word

*radius*to mean the length of this segment.For example, \(r\) is the radius of this circle with center \(O\).