Functions: From Tables to Graphs |

I know you're expecting a math lesson -- but instead, let's do some carpentry followed by a little graphic design -- activities governed by mathematics. After all, the first law of carpentry is

**measure twice -- cut once** --

that sounds like math to me.

"Carpentry??" you ask --

"Yes. We're going to build a table."

"Right! And the graphic^{ }design??"

"We'll make a picture (graph) of the data in the table."

OK? -- here goes.

Say Poor Shnook is a *glob* salesman who makes $5 for every *glob* he sells. Let's make a table of values to represent his sales *income* from the sale of 0, 1, 5, or 10 *globs*.

We'll let *x* = the number of *globs* sold and

*E*(*x*) = his *income*.

x = |
0 | 1 | 5 | 10 |

E(x) = |
$0 | $5 | $25 | $50 |

There, the carpentry's done, now for the graphic design.

Each **column** in the table is an **ordered pair **-- otherwise known as a **point**. So let's make a picture of the points (0, 0) (1, 5) (5, 25) (10, 50)

This **function** machine **transforms** a number of *globs***into** Poor Shnook's ** income**.

If he was selling his *globs* by weight at $5/Kg, then we would **let ***x*** = the weight of globs sold**. Now, we would join the points with a **continuous** line to represent the income from any weight, (say 13.45 kilos), of *globs*.

In this case, *E*(*x*) would be:

E(x) = 5*x*; *x* __>__ 0; *x* belongs to *R*

**Now, ***x* can take on any **Real Value** as long as it is positive ( __>__ 0 ).

It's against the law to sell negative quantities of *globs*.

This **function** machine **transforms** a ** weight** of

**Note:** even though the graph is a line, we refer to it as a curve. It's just a linear curve.

This is a **direct variation**** linear function**.

Its function rule is of the form ** f(x) = mx**,

** m** is the slope, and since

it passes **through the origin**.

Now, Poor Shnook gets tired of selling *globs*, so he becomes a cab driver.

He charges $2 per kilometer with a $3 pick-up-fee. (the meter starts at $3)

Let's make another table for his cabby income from a 0, 1, 5, or 10 km trip.

We let *x*** = trip distance** (in km)

so *E***(***x***) = ****income** ($)

x = |
0 | 1 | 5 | 10 |

E(x) = |
$3 | $5 | $13 | $23 |

The points indicated here are (0, 3) (1, 5) (5, 13) (10, 23) -- let's graph them.

The lesson to learn from this graph is:

Scale the *y*-axis according to the **slope of the line**, not the *y*-intercept.

It's very easy to indicate the *y*-value of the starting or **initial point**, but as we see here, it's not so easy to indicate exact values on the line when we base our scale on ** b** rather than

This is a **partial variation**** linear function**.

Its function rule is of the form ** f(x) = mx + b**, (m = 2, b = 3)

the "** mx**" is the

** m** is the slope, and since

it does **not** pass **through the origin**.

This line cuts the *y*-axis at (0, 3)

This line doesn't really reflect Poor Shnook's cab income, since the meter doesn't calculate the cost of every teeny-tiny decimal part of a kilometer. It adds a fraction of $2 every fraction of a kilometer travelled. That's exactly why **each increase is a jump** of a fixed amount instead of a constantly rising continuous increase like the line above.

Normal taxi meters tally 1/8 of a kilometer -- but this is Poor Shnook's cab -- so it only tallies every **quarter** of a kilometer -- which, at $2/km comes to $0.50 per quarter kilometer.

Let's make another table. (All this carpentry....)

We'll let ** d = distance travelled in kilometers**,

*f ***(***d***) = taxi fare in dollars**

d = |
0 | 0.25 | 0.5 | 0.75 | 1 | 1.25 |

f (d) = |
$3.00 | $3.50 | $4.00 | $4.50 | $5.00 | $5.50 |

This data or set of points describes a **Greatest Integer** (or Step) **Function**,

a **piecewise** defined function whose graph looks like steps.

Notice that *f ***(***d***) remains constant over an interval of ***d***-values** -- say from 0 to 0.25.

Since ** d** is the number of kilometers and our increase interval is every quarter of a kilometer, we'll multiply

The function rule (in Standard form) for this table's data is:

Its graph resembles steps moving upward. Each step will be ¼ of a kilometer long horizontally and the vertical separation between the steps will be 50 cents or $0.50.

For more information on this family of functions, see the lesson on the **Greatest Integer Function** in this MathRoom.

.

from data to table to graph |
practice |
solutions |

.

1) Make a table of values for each function below with *x* = 0, 1, 3 and 5.

a) f (x) = – 3x + 7 |
b) g (x) = 2 – x^{ 2} |
c) h (x) = x^{ 3} + 1 |

2) Uncle Norman rents a vehicle from "Hope-U-Make-It Car Rentals". It costs him a flat rate of $25/day with unlimited mileage. Considering the name of the rental company, he wisely buys insurance for $12 to cover the entire rental period. He then goes to visit his granny, spends two nights chez grandmere, and returns the car safely after exactly 48 hours.

- a) Write a function rule

- b) Make a table of values for

- c) Make a graph.

from data to table to graph |
practice |
solutions |

- .

1) Make a table of values for each function below with *x* = 0, 1, 3 and 5.

a) f (x) = – 3x + 7 |
b) g (x) = 2 – x^{ 2} |
c) h (x) = x^{ 3} + 1 | ||||||||||||

x |
0 | 1 | 3 | 5 | x |
0 | 1 | 3 | 5 | x |
0 | 1 | 3 | 5 |

f (x) |
7 | 4 | – 2 | – 8 | g (x) |
2 | 1 | – 7 | – 23 | h (x) |
1 | 2 | 28 | 126 |

- 2) a) Let

- b)

=d |
1 | 5 | 9 |

C() =d |
$37 | $137 | $237 |

- c)

from data to table to graph |
practice |
solutions |

(*all content **© MathRoom Learning Service; 2004 - *).