Lesson 56: Plotting Functions

New Concept

This course builds your math foundation by exploring how numbers relate to each other, using rules to solve problems and understand the world mathematically.

What’s next

To begin, we'll explore one of these core rules: functions. You will learn to represent functions with tables, equations, and graphs.

Property

The term coordinates refers to the ordered pair of numbers used to locate a point in a coordinate plane.

Examples

• The coordinates $(4, 9)$ mean start at the origin, move 4 units to the right, and then 9 units up.
• To plot the point $(-1, 6)$, you move 1 unit to the left and 6 units up from the origin.
• The point located at $(0, 0)$ is called the origin, which is the starting location for all coordinates.

Explanation

Think of coordinates as a treasure map for your graph. An ordered pair like $(x, y)$ gives you two secret directions from the origin, $(0, 0)$. The first number, $x$, tells you how far to go horizontally, and the second number, $y$, tells you how far to go vertically. Following these two steps leads you to a precise point!

Property

The rule of a function is expressed as an equation with $x$ standing for the input number and $y$ for the output number. We write the equation starting with '$y=$', such as $y = 4x$.

Examples

• A table shows pairs (1, 5), (2, 10), and (3, 15). The output is 5 times the input, so the equation is $y = 5x$.
• If a table has pairs (2, 5), (3, 6), and (4, 7), the output is always 3 more than the input, so the equation is $y = x + 3$.
• The rule 'the number of wheels is twice the number of bikes' becomes the equation $y = 2x$.

Explanation

Think of a function table as a set of clues. Your job is to be a detective and find the secret pattern connecting the input ($x$) to the output ($y$). Once you crack the code, you write it as an equation starting with '$y=$'. This equation is a powerful rule that works for any input, not just the ones in your table!

Property

If we graph all pairs for a function, the result can be an uninterrupted series of points that form a line. An arrowhead on the line shows that it continues. For example, the function $y = 2x$ forms a continuous line.

Examples

• The function $y=x+2$ includes integer points like $(1, 3)$ and $(2, 4)$, but also fractional points like $(\frac{1}{2}, 2\frac{1}{2})$.
• To graph $y=3x$, you can plot $(0, 0)$ and $(1, 3)$, then draw a straight line passing through both points.
• The graph of $y=x$ is a line that passes through $(-1, -1)$, $(0, 0)$, and $(10, 10)$, showing all points where $y$ equals $x$.

Explanation

Plotting points from a table is like putting stars in the sky. But for many functions, there are infinite points in between! We connect the dots to form a solid line when a function works for fractions and decimals, too. This line is the complete picture, showing every single input-output pair that follows the function's rule, even the tiny ones.