Comparing Functions
in Different Representations

Learn to compare key properties of functions whether they appear as equations, graphs, tables, or verbal descriptions.

Algebra 2

What Does Comparing Functions Mean?

In mathematics, a single function can be shown in four different ways: as an equation, a graph, a table of values, or a verbal description. Comparing functions means extracting the same property from two functions that may be given in different formats, then deciding which function has a larger or smaller value for that property.

The Four Representations

Algebraic

An equation like $ f(x) = -2x^2 + 8x - 3 $. You extract properties by computation.

Graphical

A plotted curve. You read properties visually: intercepts, peaks, end behavior.

Numerical (Table)

A set of $ (x, y) $ pairs. You scan for patterns, max/min values, and rate of change.

Verbal

A word description: "The profit starts at $200 and grows by 15% each year."

Why This Skill Matters

Real-world data rarely arrives in one neat format. A scientist might have experimental data in a table and a theoretical model as an equation. To compare them, you must extract the same property from each format.

1

Flexibility: You can work with whatever format is given.

2

Deeper understanding: Connecting representations reveals why functions behave as they do.

3

Problem solving: Many standardized test and AP questions mix representations deliberately.

Key Properties to Compare

When comparing two functions, focus on these core properties. Each can be found from any representation:

Intercepts

$ x $-intercepts (zeros) and $ y $-intercept $ f(0) $.

Maximum / Minimum

The vertex of a parabola, or the peak/trough of a curve.

Rate of Change

Slope (linear), average rate between points, or growth factor.

Domain & Range

What $ x $-values are valid and what $ y $-values are achieved.

End Behavior

What happens as $ x \to \infty $ and $ x \to -\infty $.

Increasing / Decreasing

Intervals where the function rises or falls.

Core Rule: Extract, Then Compare

The key insight: the property is the same regardless of the representation. A maximum is a maximum whether you find it from a vertex formula, read it off a graph, or identify it in a table.

Extracting Properties: A Quick Guide

Property	Algebraic	Graph	Table
$ y $-intercept	Evaluate $ f(0) $	Read where curve crosses $ y $-axis	Find row where $ x = 0 $
Maximum	Vertex: $ x = \frac{-b}{2a} $	Read highest point	Largest $ y $-value in table
Zeros	Solve $ f(x) = 0 $	$ x $-intercepts	Rows where $ y = 0 $
Rate of change	Slope $ m $, or $ \frac{\Delta y}{\Delta x} $	Steepness of curve	$ \frac{y_2 - y_1}{x_2 - x_1} $

Exact vs. Approximate Values

Different representations give different levels of precision. Understanding this distinction prevents errors:

Exact

Algebraic representations give exact values. Computing $ f(0) = -3 $ is precise.

Approximate

Graphs require reading coordinates, which can be imprecise. A peak might look like $ y \approx 5 $ but actually be $ y = 5.25 $.

How to Apply: Step-by-Step

1

Read the question carefully. What property are you comparing? (maximum, intercept, rate of change, etc.)

2

Identify each function's format. Is it an equation? A graph? A table? A description?

3

Extract the property from Function A using the method suited to its format.

4

Extract the same property from Function B using the method suited to its format.

5

Compare the two values and write your conclusion: which is larger, smaller, or are they equal?

Example 1: Comparing Maximums

Function A (equation): $ f(x) = -x^2 + 6x - 4 $
Function B (graph below): Which has the larger maximum?

Example 2: Table vs. Equation

Function A (table):

$ x $	$ h(x) $
0	3
1	6
2	12
3	24
4	48

Function B (equation): $ k(x) = 5 \cdot 3^x $

Which function has the greater $ y $-intercept? Which grows faster?

Example 3: Graph vs. Verbal

Function A (graph below): a linear function.
Function B (verbal): "A plumber charges a $40 service fee plus $25 per hour."
Which has the greater rate of change?

Example 4: Two Quadratics, Two Formats

Function A (equation): $ f(x) = 2x^2 - 8x + 10 $
Function B (table):

$ x $	-1	0	1	2	3	4
$ g(x) $	10	5	2	1	2	5

Which has the smaller minimum value?

Common Mistakes

✕ Wrong Property

Comparing the $ y $-intercept of one function with the maximum of another. Always extract the same property from both.

✕ Misreading the Graph

Estimating a vertex at $ y = 4 $ when it is actually at $ y = 5 $. Use gridlines carefully.

✕ Table Gaps

Assuming the largest $ y $ in the table is the true max. The vertex might lie between listed $ x $-values.

Summary: Representation Strengths

Representation	Strengths	Limitations
Algebraic	Exact values, any $ x $	Hard to "see" overall shape
Graphical	Visual shape, end behavior	Imprecise readings
Table	Exact at listed points	May miss values between rows
Verbal	Real-world context	Must translate to math first

Question 1 of 5

Function $ f $ is given by $ f(x) = -3x^2 + 12x - 7 $. Function $ g $ has a graph with a vertex at $ (3, 6) $ opening downward. Which function has the larger maximum?

For $ f $: vertex at $ x = \frac{-12}{2(-3)} = 2 $, so $ f(2) = -12 + 24 - 7 = 5 $. The graph tells us $ g $ peaks at $ 6 $. Since $ 6 > 5 $, $ g $ has the larger maximum.

Question 2 of 5

A table for function $ p $ shows: $ p(0) = 2 $, $ p(1) = 6 $, $ p(2) = 18 $, $ p(3) = 54 $. Function $ q $ is described as: "starts at 5 and triples every unit." Which has the greater $ y $-intercept?

The $ y $-intercept is the value when $ x = 0 $. From the table, $ p(0) = 2 $. The verbal description says $ q $ "starts at 5," so $ q(0) = 5 $. Since $ 5 > 2 $, $ q $ has the greater $ y $-intercept. Note that both functions triple each step, so they grow at the same rate: the difference is only in the starting value.

Question 3 of 5

Function $ f(x) = 4x + 3 $. A graph of function $ g $ passes through $ (0, 7) $ and $ (2, 13) $. Which function has the greater rate of change?

For $ f $, the slope is the coefficient of $ x $: 4. For $ g $, the slope is $ \frac{13 - 7}{2 - 0} = \frac{6}{2} = $$3$. Since $ 4 > 3 $, $ f $ has the greater rate of change. The $ y $-intercept of $ g $ is larger, but that is a different property than rate of change.

Question 4 of 5

A table for $ m(x) $ is shown: $ m(-1) = 8,\; m(0) = 5,\; m(1) = 4,\; m(2) = 5,\; m(3) = 8 $. The equation $ n(x) = x^2 - 2x + 3 $ defines another function. Which has the smaller minimum?

From the table, the smallest value is $ m(1) = 4 $. For $ n(x) = x^2 - 2x + 3 $: vertex at $ x = \frac{2}{2} = 1 $, so $ n(1) = 1 - 2 + 3 = $$2$. Since $ 2 < 4 $, $ n $ has the smaller minimum.

Question 5 of 5

Function $ f $ is described verbally: "A ball is thrown upward from 6 feet, reaches a maximum height of 22 feet, and lands after 3 seconds." Function $ g(t) = -16t^2 + 48t + 4 $. Which has the greater maximum height?

The verbal description states $ f $ reaches 22 ft. For $ g(t) = -16t^2 + 48t + 4 $: vertex at $ t = \frac{-48}{2(-16)} = 1.5 $. Then $ g(1.5) = -16(2.25) + 48(1.5) + 4 = -36 + 72 + 4 = $$40$. Since $ 40 > 22 $, $ g $ has the greater maximum height.

Property	Algebraic	Graph	Table
\( y \)-intercept	Evaluate \( f(0) \)	Read where curve crosses \( y \)-axis	Find row where \( x = 0 \)
Maximum	Vertex: \( x = \frac{-b}{2a} \)	Read highest point	Largest \( y \)-value in table
Zeros	Solve \( f(x) = 0 \)	\( x \)-intercepts	Rows where \( y = 0 \)
Rate of change	Slope \( m \), or \( \frac{\Delta y}{\Delta x} \)	Steepness of curve	\( \frac{y_2 - y_1}{x_2 - x_1} \)

Representation	Strengths	Limitations
Algebraic	Exact values, any \( x \)	Hard to "see" overall shape
Graphical	Visual shape, end behavior	Imprecise readings
Table	Exact at listed points	May miss values between rows
Verbal	Real-world context	Must translate to math first

Comparing Functionsin Different Representations

What Does Comparing Functions Mean?

The Four Representations

Why This Skill Matters

Key Properties to Compare

Core Rule: Extract, Then Compare

Extracting Properties: A Quick Guide

Exact vs. Approximate Values

How to Apply: Step-by-Step

Example 1: Comparing Maximums

Example 2: Table vs. Equation

Example 3: Graph vs. Verbal

Example 4: Two Quadratics, Two Formats

Common Mistakes

Summary: Representation Strengths

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Comparing Functions
in Different Representations