Lesson 47: Graphing Functions

New Concept

A function is a rule that pairs one output number with each input number. We can represent these special input-output relationships in four ways: descriptions, equations, tables, or graphs.

What’s next

Let's put this concept into action. You'll soon build function tables from rules and create graphs for different scenarios, even one that looks like a set of stairs!

Property

A function is a rule that pairs one output number with each input number. The set of inputs is called the domain, and the set of outputs is called the range.

Examples

• In the function $y = x + 10$, if the input is $x=5$, the only possible output is $y=15$.
• A recipe function states that for every 2 cups of flour (input), you get 12 cookies (output).
• A function machine with the rule $y = 3x$ turns an input of $x=4$ into an output of $y=12$.

Explanation

Think of a function as a trusty vending machine! For every specific button you press (the input), you get one, and only one, specific snack (the output). It's a reliable rule that prevents surprises. A checkout scanner does this too: one barcode always pulls up the same price, ensuring every item has its assigned cost without any guesswork.

Property

A function whose graph is composed of a series of horizontal line segments, creating a stair-step pattern. For any time within an interval the value is constant, but it jumps to a new value at the start of the next interval.

Examples

• A parking garage charges 2 dollars for the first hour and jumps to 4 dollars for any time over 1 hour up to 2 hours.
• A library charges a late fee of 1 dollar per week. A book 1 to 7 days late costs 1 dollar; 8 to 14 days late costs 2 dollars.
• A phone plan costs 25 dollars for up to 5 GB of data, then jumps to 35 dollars the moment you exceed 5 GB.

Explanation

Imagine you're playing a video game where you level up! You stay at Level 1 for a while, and then BAM, you instantly jump to Level 2. A step function works just like that. The output value stays flat for a range of inputs, then suddenly leaps to a new level, making a graph that looks exactly like a staircase.

Property

A function that can be graphed as a single, uninterrupted line or curve. This means there are no gaps, jumps, or holes in the graph.

Examples

• The function $P = 4s$, for the perimeter of a square, is continuous because side length s can be any positive value.
• A graph showing a person's height over time would be continuous, as growth is a steady process.
• The function $y=2x$ is continuous because for every value of $x$, including fractions, there is a corresponding value of $y$ on a single line.

Explanation

Think of a continuous function like a smoothly flowing river. There are no sudden stops or teleporting water; it just flows without any breaks. The graph of a continuous function is the same way—you can draw the entire thing without ever lifting your pencil off the paper. It represents a gradual and unbroken change between input and output values.