Lesson 5-4: Compare Functions

Property

A single mathematical relationship can be represented in four connected ways:

1. Word Problem: Describes the real-world scenario.
2. Equation: States the algebraic rule (e.g., y = mx + b).
3. Table: Lists specific input-output values (ordered pairs).
4. Graph: Visually plots the ordered pairs to show the pattern.

Examples

Let's represent one situation in all four ways:

• Word Problem: A student reads 10 pages of a book every day.
• Equation: p = 10d (where d is days and p is total pages).
• Table:

• Graph: You would label the x-axis "Days" and the y-axis "Pages", plot the points (1, 10), (2, 20), and (3, 30), and draw a line through them starting from (0,0).

Explanation

Equations, tables, graphs, and word problems are just different languages telling the exact same mathematical story! If you have an equation, you can build a table. If you have a table, you can draw a graph. If you look at a graph, you can figure out the equation. Mastering how to translate between these four representations is the ultimate key to understanding algebra.

Property

The initial value (or y-intercept, b) is the starting point of a function, which is the output value when the input x = 0.

• From a Graph: It is the y-coordinate of the point where the line crosses the y-axis.
• From a Table: If x = 0 is not listed, find the rate of change (m) and work backward, or substitute a point (x, y) into y = mx + b to solve for b.

Examples

• Comparing Equations: Compare y = 2x + 5 and y = 3x + 1. The first function starts at (0, 5) while the second starts at (0, 1), so the first function has a higher initial value.

• The Table Trap: A table shows points (2, 10) and (4, 16). The rate of change is m = 3. The first y-value is 10, but since x = 2, 10 is NOT the initial value. Working backward to x = 0, subtract 2 times the rate of change: 10 - 2(3) = 4. The true initial value is 4.

Property

The rate of change (or slope, m) describes how much the output (y) changes for every one-unit increase in the input (x).

• From a Graph: Calculate the "rise over run" between any two points.
• Comparing Slopes: The absolute value of the slope indicates how quickly the function values change. A larger absolute value means a steeper line and a faster rate of change.

Examples

• Comparing Speeds: Compare f(x) = 2x + 1 and g(x) = 1/3x + 1. Since 2 is greater than 1/3, function f changes more rapidly than function g.
• Comparing Decreases: Compare h(x) = -3x + 5 and k(x) = -x + 5. Since the absolute value of -3 is greater than the absolute value of -1, function h decreases more rapidly than function k.

Explanation

The slope is the "speed limit" of your function. It tells you the rate at which a quantity is changing, such as cost per hour or distance per minute. When comparing two functions, the one with the steeper slope (larger absolute value, whether positive or negative) is the one changing the fastest.