Comparing Functions

A cell phone plan charges a flat fee of $20 per month plus $0.05 per text message. A competing plan is shown in the table below, listing total monthly cost against number of texts sent. A customer wants to know which plan costs less for heavy texters. To answer that, you need to compare two functions that are not even presented in the same form — one is a formula, the other is a set of data points. That is the core challenge this lesson addresses.

The key idea is this: a function's properties — its slope, its intercepts, its maximum or minimum, its end behavior — can be read from any representation. A graph, a table, an equation, and a verbal description can all encode the same information. Your job is to extract properties from each form and put them on equal footing.

The most common properties you will compare are the slope (rate of change), the y-intercept (starting value), maximum or minimum values, and which function produces larger outputs over a given interval.

Start with the algebraic function. If you have something like $f(x) = 3x + 7$, you can read the slope directly as 3 and the y-intercept directly as 7. For a quadratic like $g(x) = -2(x-1)^2 + 8$, the vertex is at $(1, 8)$ and the parabola opens downward, so the maximum value is 8. These facts come straight from the equation.

Now look at a function given as a graph or table. The y-intercept is wherever the graph crosses the vertical axis — or in a table, the output when the input is 0. The slope of a linear function from a table is the change in output divided by the change in input between any two rows. For a quadratic from a graph, the vertex is the highest or lowest point you can see.

The comparison only works when you are measuring the same property. If you want to compare slopes, extract the slope from both functions. If you want to compare maximum values, find the maximum of both. Mixing up which property you computed for which function is the most common error on this topic.

Here is a direct comparison between a function given algebraically and one given as a table.

Let $f(x) = 2x + 1$ and let $g$ be defined by the table:

| $x$ | $g(x)$ | |---|---| | 0 | 3 | | 1 | 6 | | 2 | 9 | | 3 | 12 |

Compare the slopes and y-intercepts of the two functions.

Now try a comparison involving a quadratic. This requires finding a vertex from one representation and matching it against something read from a graph.

Let $f(x) = -(x-2)^2 + 9$ and let $g$ be the function shown in the graph below. Both open downward. Which has the greater maximum value?

The red curve is $f(x) = -(x-2)^2 + 9$. The blue curve represents $g$, which in this case has the equation $g(x) = -(x+1)^2 + 4$ but is being treated as if it were only given graphically. Read $g$'s maximum from the graph; find $f$'s maximum from the formula.

One more situation comes up frequently: comparing an algebraic function to a verbal description. Suppose someone tells you that function $h$ starts at a value of 5 when $x = 0$ and increases by 4 for every unit increase in $x$. You need to compare that to a given formula. The verbal description is just a slope and a y-intercept stated in plain English — translate it and then compare directly.

The verbal description above translates to $h(x) = 4x + 5$. If the comparison function is $k(x) = 6x + 2$, then $k$ has a greater slope but a smaller y-intercept. For small values of $x$, $h$ will be larger. For large values of $x$, $k$ will dominate. Finding exactly where they cross requires setting them equal and solving, which is a tool from systems of equations — but recognizing that both effects exist is the comparison skill itself.