Home › Functions and graphs

How to read a cost graph: C(0), C(5), C(20) and the formula behind it

The rule: the input is on the horizontal axis, the output on the vertical one

In a notation like $C (20)$ the number in brackets is always the input - the quantity on the horizontal axis - and the result is the output from the vertical axis. Mixing them up is the usual mistake: $C (5) = 40$ means "five gigabytes cost forty dollars", never "five dollars buy forty gigabytes". To interpret a value in words, read the axis labels literally and put them into a sentence: input units in, output units out. A straight stretch has a constant rate of change, the slope:

rate = \frac{change in output}{change in input}

and its units are output-per-input - here dollars per gigabyte. A flat stretch means the output does not depend on the input at all: on a bill that is a base fee.

Problem

A graph shows the monthly cost $C$ in dollars against the number of gigabytes $g$ used. It consists of a horizontal segment from $(0, 40)$ to $(5, 40)$ and a straight climb from $(5, 40)$ to $(30, 290)$ , passing through $(20, 190)$ . (a) Find $C (0)$ and interpret it. (b) Find $C (5)$ and interpret it. (c) Find $C (20)$ and interpret it. (d) What is the cost of each gigabyte above the first five? (e) Write $C (g)$ as a formula and state the domain.

The plan: a flat $40 for the first 5 GB, then $10 for every extra gigabyte.

Answer

C (0) = 40

dollars - the base fee paid before any data is used.

C (5) = 40

dollars - five gigabytes are still covered by that fee.

C (20) = 190

dollars. Above 5 GB the rate is

10

dollars per gigabyte, so

C (g) = 40

for

0 \leq g \leq 5

and

C (g) = 10 g - 10

for

5 < g \leq 30

. Domain:

[0, 30]

Checked by hand: the rate and the piecewise formula reproduce all three marked points.

Step-by-step solution

(a) C(0) - the base fee

Go to $g = 0$ on the horizontal axis and read the height: the graph starts at $(0, 40)$ , so

C (0) = 40

In words: before a single gigabyte is used the plan already costs $40$ dollars a month - a base fee.

(b) C(5) - still the same forty

The first stretch is flat, so nothing changes up to five gigabytes:

C (5) = 40

In words: if $5$ gigabytes are used, the monthly charge is $40$ dollars. Note what the sentence does not say - it is not "five dollars" and not "forty gigabytes". The bracketed number is gigabytes, the answer is dollars.

(c) C(20)

The marked point on the climbing part is $(20, 190)$ , so

C (20) = 190

In words: using $20$ gigabytes brings the bill to $190$ dollars.

(d) The price of one extra gigabyte

On the straight climb the rate is the same everywhere, so take its two ends:

rate = \frac{290 - 40}{30 - 5} = \frac{250}{25} = 10

Each gigabyte above the first five costs $10$ dollars. Check it against the middle point: $40 + 10 \cdot (20 - 5) = 40 + 150 = 190$ - exactly $C (20)$ .

(e) The formula

Two stretches mean two lines, so the function is piecewise:

C (g) = {40, 40 + 10 (g - 5) = 10 g - 10, 0 \leq g \leq 5 5 < g \leq 30

The graph exists only between $g = 0$ and $g = 30$ , so

D : [0, 30]

In practice that means the plan simply does not offer more than $30$ gigabytes.

How to check yourself

Substitute the marked points back into the formula. $C (5) = 40$ from the first line and $10 \cdot 5 - 10 = 40$ from the second - they agree at the joint, which is what makes the graph unbroken. And $C (30) = 10 \cdot 30 - 10 = 290$ , the right end of the picture.

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.