Investigation 8: Analyzing and Graphing Relationships

New Concept

Now we will learn how to write an equation to represent the relationship in the table.

What’s next

Next, you’ll analyze relationships in tables, plot them on coordinate graphs, and write equations to represent the patterns you find.

An equation describes the consistent relationship between two sets of data in a table. By analyzing pairs of values, we can find a multiplication formula that connects them. This rule allows you to predict outcomes for any input, like calculating a quiz score or total pay based on a single number, which makes analyzing data much faster.

Quiz Table: If 4 correct answers give a score of 40%, the rule is $\text{Score} = \text{Correct Answers} \times 10$.
Pay Table: If 3 hours of work earns 30 dollars, the rule is $\text{Pay} = \text{Hours Worked} \times 10$.
Hiking Table: If 2 hours of hiking covers 8 miles, the rule is $\text{Miles} = \text{Hours} \times 4$.

Think of yourself as a data detective! Your mission is to find the single 'magic number' that consistently turns the first column's value into the second. This constant multiplier is the key that unlocks the equation and reveals the hidden pattern in the numbers.

Graphs provide a visual way to see the relationship between two quantities. Data from a table, like hours worked and pay earned, can be plotted as points on a coordinate plane. The position of each dot represents a pair of values. Connecting these dots with a line can help us estimate values that fall between the given points.

From a pay table, the entry for 4 hours and 40 dollars is plotted as the single point $(4, 40)$ on the graph.
Rosita hiked for 3 hours and went 12 miles. This relationship is shown by a dot at the coordinates $(3, 12)$.
By drawing a line through points $(1,4)$ and $(2,8)$, we can find the point for $1\frac{1}{2}$ hours, which is at 6 miles.

Turn your data table into a picture! Each row is an instruction for placing a dot. The horizontal value tells you how far to walk right, and the vertical value tells you how high to climb up. This visual map makes understanding the data relationship a breeze!

Coordinates are pairs of numbers used to name points on a grid, written in parentheses like $(x, y)$. The first number is taken from the horizontal scale (x-axis), and the second number is from the vertical scale (y-axis). The point where the axes meet is the origin, $(0,0)$, which is the starting point for all plotting.

The coordinates $(7, 8)$ mean: start at the origin, move 7 units right along the horizontal axis, then 8 units up.
Point A at $(7, 8)$ and Point B at $(7, 3)$ share the same horizontal position but have different vertical positions.
To plot point C at $(1, 3)$, you first move 1 unit to the right and then 3 units up from the origin.

Coordinates are a point's secret address on a grid. Just remember to walk along the horizontal hallway first to find your number, and then take the vertical elevator up to find its partner. Always go horizontal, then vertical, to pinpoint the exact location every time!