Lesson 3Equations for Functions

Learning Goal

Let’s find outputs from equations.

Learning Targets

  • I can find the output of a function when I know the input.

  • I can name the independent and dependent variables for a given function and represent the function with an equation.

Lesson Terms

  • dependent variable
  • independent variable
  • radius

Warm Up: A Square’s Area

Problem 1

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 is side length of square for values: 8, 2,2, 12 and one-fourth, s. Output is A, the area of the square

input

output

Activity 1: Diagrams, Equations, and Descriptions

Problem 1

Record your answers to these questions in the table provided:

  1. Match each of these descriptions with a diagram:

    1. the circumference, , of a circle with radius,

    2. the distance in miles, , that you would travel in hours if you drive at 60 miles per hour

    3. the output when you triple the input and subtract 4

    4. the volume of a cube, , given its edge length,

  2. Write an equation for each description that expresses the output as a function of the input.

  3. Find the output when the input is 5 for each equation.

  4. Name the independent and dependent variables of each equation.

A: input s, rule is s cubed, output v. B: input t, rule 60t, output d. C: input x, rule is 3x - 4, output y. D: input is r, rule is (2)(pi)(r), output is C

description

a

b

c

d

diagram

equation

input = 5
output = ?

independent
variable

dependent
variable

Are you ready for more?

Problem 1

Choose a 3-digit number as an input and 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?

Activity 2: Dimes and Quarters

Problem 1

Jada had some dimes and quarters that had a total value of $12.50. The relationship between the number of dimes, , and the number of quarters, , can be expressed by the equation .

  1. If Jada has 4 quarters, how many dimes does she have?

  2. If Jada has 10 quarters, how many dimes does she have?

  3. Is the number of dimes a function of the number of quarters? If yes, write a rule (that starts with  …) that you can use to determine the output, , from a given input, . If no, explain why not.

  4. If Jada has 25 dimes, how many quarters does she have?

  5. If Jada has 30 dimes, how many quarters does she have?

  6. Is the number of quarters a function of the number of dimes? If yes, write a rule (that starts with  …) that you can use to determine the output, , from a given input, . If no, explain why not.

Lesson Summary

We can sometimes represent functions with equations. For example, the area, , of a circle is a function of the radius, , and we can express this with an equation:

We can also draw a diagram to represent this function:

Input is r, rule is (pi)(r squared), output is A

In this case, we think of the radius, , as the input, and the area of the circle, , as the output. For example, if the input is a radius of 10 cm, then the output is an area of cm², or about 314 square cm. Because this is a function, we can find the area, , for any given radius, .

Since it is the input, we say that is the independent variable and, as the output, 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

then we can think of as a function of and write

or we can think of as a function of and write