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.

"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) = 5x; 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 globs into Poor Shnook's income.

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 b = 0,

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 m.

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 part that varies -- the b is constant.

m is the slope, and since b is not = 0,

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 d by 4 to have the number of quarter kilometers. Then we'll multiply that value by \$0.50 -- the price per quarter kilometer.

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.

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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 C(d) which calculates Uncle Norman's bill for the rental car if he rents the car for d days.
b) Make a table of values for C(d) for d = 1, 5, 9 days.
c) Make a graph.
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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 d = the number of days the car is rented
C(d) = 25d + 12; d > 0, C(d) in \$.
b)
 d = 1 5 9 C(d) = \$37 \$137 \$237
c) (all content © MathRoom Learning Service; 2004 - ).