SAT — mathHard6 min read

Comparing Linear Functions

Compare linear functions given in different forms for the Digital SAT. Analyse equations, tables, and graphs to determine which function is greater.

Tutor AI Team2026-02-25T01:13:42.559Z

The Digital SAT often presents two or more linear functions in different representations — one as an equation, another as a table, and a third as a graph — and asks you to compare them. Which has the greater slope? Which has the greater y-intercept? At what input do they produce the same output?

This question type tests your ability to extract the same information (slope, y-intercept, specific values) regardless of how the function is presented.

Core Concepts

Extracting Slope and Y-Intercept from Different Representations

From an equation $f(x) = mx + b$ :

Slope = $m$ , y-intercept = $b$

From a table:

Slope = $\frac{\Delta y}{\Delta x}$ (change in output ÷ change in input)
Y-intercept: find the row where $x = 0$ , or calculate using $b = y - mx$

From a graph:

Slope: pick two points and calculate rise/run
Y-intercept: where the line crosses the y-axis

Comparing Slopes

The slope tells you the rate of change. A function with a steeper line (larger $|m|$ ) changes faster. Compare slopes to determine:

Which function increases/decreases more quickly
Whether one function will eventually overtake the other

Example: $f(x) = 3x + 1$ grows faster than $g(x) = 2x + 5$ because $3 > 2$ .

Comparing Y-Intercepts

The y-intercept tells you the starting value. The function with the larger y-intercept starts higher.

Finding Where Two Functions Are Equal

Set the functions equal: $f(x) = g(x)$ and solve for $x$ .

Example: $f(x) = 3x + 1$ and $g(x) = 2x + 5$

$3x + 1 = 2x + 5$ → $x = 4$

At $x = 4$ , both functions equal 13.

Determining Which Function Is Greater

For $x < 4$ : $g(x) > f(x)$ (higher y-intercept wins initially)
For $x > 4$ : $f(x) > g(x)$ (steeper slope catches up and overtakes)
At $x = 4$ : $f(x) = g(x)$

Average Rate of Change

For a linear function, the average rate of change over any interval equals the slope. This is a common SAT question:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a} = m$

Strategy Tips

Tip 1: Convert Everything to the Same Form

The easiest comparison is when all functions are in slope-intercept form. Convert tables and graphs to equations first.

Tip 2: Use Specific Values to Compare

If you're stuck, evaluate both functions at the same $x$ -value to see which gives a larger output.

Tip 3: Be Careful with Negative Slopes

A slope of $-5$ is steeper than $-2$ in terms of absolute value, but it represents a faster decrease.

Tip 4: Read Graph Scales Carefully

Don't assume each grid square is 1 unit. Check the axis labels.

Tip 5: Average Rate of Change = Slope

For any linear function, the average rate of change between any two points is always the slope. This is a shortcut on the SAT.

Worked Example: Example 1

Problem

Function $f$ is defined by $f(x) = 4x - 3$ . Function $g$ is shown in the table below. Which function has the greater slope?

$x$	$g(x)$
0	2
2	12
4	22

Slope of $f$ : $4$

Slope of $g$ : $\frac{12 - 2}{2 - 0} = 5$

$g$ has the greater slope.

Solution

Worked Example: Example 2

Problem

$f(x) = -2x + 10$ and $g(x) = x + 1$ . For what value of $x$ is $f(x) = g(x)$ ?

$-2x + 10 = x + 1$

$9 = 3x$

$x = 3$

Solution

Worked Example: Example 3

Problem

Function $f$ has a y-intercept of 8 and passes through $(4, 20)$ . Function $g(x) = 2x + 10$ . Which function has the greater value at $x = 6$ ?

Slope of $f$ : $\frac{20 - 8}{4 - 0} = 3$

$f(x) = 3x + 8$ → $f(6) = 26$

$g(6) = 2(6) + 10 = 22$

$f(6) > g(6)$ : $f$ has the greater value at $x = 6$ .

Solution

Worked Example: SAT-Style

Problem

The average rate of change of function $h$ over the interval $[2, 6]$ is 5. If $h(2) = 3$ , what is $h(6)$ ?

Average rate of change $= \frac{h(6) - h(2)}{6 - 2} = 5$

$\frac{h(6) - 3}{4} = 5$

$h(6) - 3 = 20$

$h(6) = 23$

Solution

Worked Example: Example 5

Problem

Function $p(x) = ax + b$ passes through $(1, 5)$ and $(3, 11)$ . Function $q(x) = cx + d$ passes through $(0, 8)$ and $(2, 12)$ . Compare their slopes.

Slope of $p$ : $\frac{11 - 5}{3 - 1} = 3$

Slope of $q$ : $\frac{12 - 8}{2 - 0} = 2$

$p$ has a steeper slope ( $3 > 2$ ).

Solution

Practice Problems

Problem 1

$f(x) = 5x + 2$ . Table for $g$ : $(1, 10)$ , $(3, 18)$ , $(5, 26)$ . Which has the greater slope?

Problem 2

$f(x) = -x + 12$ and $g(x) = 2x$ . Find where $f(x) = g(x)$ and which is greater at $x = 5$ .

Problem 3

Function $h$ has average rate of change 3 on $[0, 4]$ and $h(0) = 7$ . Find $h(4)$ .

Problem 4

$f(x) = 2x + 1$ and $g(x) = -3x + 16$ . For what $x$ does $f(x) = g(x)$ ? For $x > 3$ , which function is larger?

Problem 5

A graph shows a line through $(0, 5)$ and $(3, 11)$ . An equation states $g(x) = 3x + 1$ . Compare slopes and y-intercepts.

Problem 6

The average rate of change of $f$ from $x = 1$ to $x = 5$ is $-2$ . If $f(1) = 10$ , what is $f(5)$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Comparing absolute values instead of actual values. A slope of $-5$ is less than 2, not greater, even though $|-5| > |2|$ .
Misreading table values. Double-check which column is $x$ and which is $f(x)$ .
Assuming same scale on graph axes. Always check the labels.
Forgetting that average rate of change = slope for linear functions. This simplification saves time.
Not finding the y-intercept from a table. If $x = 0$ isn't in the table, calculate $b$ using slope and a known point.

Frequently Asked Questions

What if the functions are presented in different units?

You can't directly compare slopes if the units differ. The SAT ensures units match when asking for comparisons.

Can I use a calculator to compare functions?

Yes — graph both functions on the Desmos calculator to visually compare slopes, intercepts, and intersection points.

What does "greater rate of change" mean?

It means a steeper slope in the positive direction. Technically, 5 > 3 > 0 > -2 > -5 for rate of change.

How do I compare a decreasing function with an increasing function?

The increasing function always has the greater rate of change (positive > negative). For specific values, evaluate both at the given $x$ .

Is average rate of change always the slope?

For linear functions, yes. For nonlinear functions, the average rate of change varies by interval.

Key Takeaways

✓
Extract slope and y-intercept from equations, tables, and graphs using the same techniques.
✓
Set functions equal to find where they intersect.
✓
Greater slope means faster growth (for positive slopes) or slower decline.
✓
Average rate of change = slope for linear functions.
✓
Convert to a common form (slope-intercept) for the easiest comparison.
✓
Use the Desmos calculator on the Digital SAT for visual confirmation.

Tags:

#sat#math#algebra#functions#comparing

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Core Concepts

Extracting Slope and Y-Intercept from Different Representations

Comparing Slopes

Comparing Y-Intercepts

Finding Where Two Functions Are Equal

Determining Which Function Is Greater

Average Rate of Change

Strategy Tips

Tip 1: Convert Everything to the Same Form

Tip 2: Use Specific Values to Compare

Tip 3: Be Careful with Negative Slopes

Tip 4: Read Graph Scales Carefully

Tip 5: Average Rate of Change = Slope

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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