Comparing Functions Across Tables, Graphs and Equations

One function, four appearances

The same function can be written four ways.

As an equation:

$$y = 3x + 5$$

As a table:

$x$	$0$	$1$	$2$	$3$
$y$	$5$	$8$	$11$	$14$

As a graph: a line crossing the $y$-axis at $5$ and rising $3$ for every $1$ across.

In words: start at $5$ and add $3$ each time.

None is more correct than the others. Which one you meet depends on where the information came from — an experiment produces a table, a formula produces an equation, a report produces a graph.

Two numbers settle a linear function

$$y = mx + b$$

Number	Name	What it is
$m$	rate of change	how much $y$ changes per step of $x$
$b$	starting value	the $y$ when $x = 0$

Every linear function is determined by those two. So comparing two functions shown in different forms reduces to finding $m$ and $b$ in each and comparing them — no conversion to a common format is needed.

Finding them in each form

From an equation: read them off. In $y = 3x + 5$, $m = 3$ and $b = 5$.

From a graph: $b$ is where the line crosses the $y$-axis, and $m$ is the slope — rise over run.

From a table: divide the change in $y$ by the change in $x$.

$x$	$2$	$5$	$8$
$y$	$11$	$20$	$29$

$$m = \frac{20 - 11}{5 - 2} = \frac{9}{3} = 3$$

The table has no $x = 0$ row, so step backwards from $(2, 11)$ by two steps of $3$:

$$b = 11 - 2(3) = 5$$ $$y = 3x + 5$$

From a description: the amount that repeats per unit is $m$; the amount charged once is $b$.

Comparing two functions

Function A: $y = 4x + 10$.

Function B:

$x$	$0$	$1$	$2$	$3$
$y$	$25$	$28$	$31$	$34$

Extract both pairs:

	Rate of change	Starting value
A	$4$	$10$
B	$3$	$25$

B starts higher; A grows faster. Those two facts answer most questions that get asked about a pair like this, including the one worth asking next: A eventually overtakes B, and you can find where by setting them equal.

$$4x + 10 = 3x + 25 \quad\Rightarrow\quad x = 15$$

At $x = 15$ both give $70$. Before that B leads; after it, A does.

Building a function from a description

The wording tells you which number is which.

A gym charges $30$ to join plus $12$ a month.

The $12$ repeats every month, so it attaches to the variable. The $30$ happens once, so it stands alone:

$$y = 12x + 30$$

A tank holds $200$ liters and drains at $15$ liters a minute.

Draining makes the rate negative:

$$y = 200 - 15x$$

Phrase	Points to
per, each, every, a month	the rate of change
fee, deposit, starts at, already has	the starting value

Why comparing across forms is the useful skill

Real information does not arrive in a chosen format. A phone plan is described in words, a bill arrives as a table, a report shows a graph — and the question is often which of two options is better.

Converting everything into one form is slow and adds chances to slip. Pulling the same two numbers out of each is quick, and it makes the comparison structural: which grows faster, and which starts higher. Those two facts also decide whether one option ever overtakes the other, and when.

Beyond straight lines

A function that is not linear is not settled by a rate and a start, but the method survives. Decide which feature the question is about, then pull that feature out of each form.

Type	Features that settle a comparison
linear	rate of change, starting value
quadratic	the vertex — a maximum or minimum — and the zeros
exponential	starting value, growth factor

Take two quadratics. Function A is known only from its graph:

• Function A

Function B is known only from its rule:

$$B(x) = -2x^2 + 8x + 15$$

Which has the larger maximum? Function A’s maximum is read off the graph: its highest point is $(3, 20)$, so the maximum is $20$.

Function B’s maximum sits at its vertex. The axis is at $x = -\tfrac{b}{2a} = -\tfrac{8}{2(-2)} = 2$.

$$B(2) = -2(4) + 8(2) + 15 = -8 + 16 + 15 = 23$$

B has the larger maximum, $23$ against $20$ — even though A’s peak comes later. Neither function had to be converted into the other’s form; each one gave up its vertex in the way its own form allows.

Worked examples

Common mistakes

Practice problems