Drawing the Graph

The previous page said the graph is the record of every pair the function produces. Every pair is infinitely many of them, and you have a pencil and an afternoon. So nobody draws a graph that way. What people do instead is pick a short list of inputs, work out the output for each one, mark the pairs, and join them up.

The list is a table of values, and it is the graph filed differently. One row holds one input and its output. That row is one point. Nothing is added when the row becomes a dot and nothing is lost, which is the same claim the fold made on the previous page, now with the pairs arriving one at a time instead of all at once.

Choosing the inputs is the part with judgement in it. Whole numbers spread evenly are the usual opening, because they are quick to work out and they cover ground. Then look at what came back. Where three dots in a row sit almost on a straight line, a fourth between them tells you little. Where the dots change direction, crowd together, or leave a gap, that is where another input earns its place.

Joining the dots is a separate act and it is not part of the record. The rule was asked about the inputs in your table and about no others. The line you draw between two dots is your claim about what the rule does in between, and the rule was never consulted. Usually that claim is close to true. The next step is about the times it is not.

Every graph you have seen on a screen was made the way you made one in the last step. A plotting program picks a number of x values across the window, works out the rule at each, and joins consecutive results with straight segments. What is drawn is a chain of straight pieces, a polyline, with the pieces short enough that it reads as a curve. matplotlib does this. So does a graphing calculator.

Sampling Working out a function at a finite list of inputs and treating the results as a stand-in for the whole thing. Every plotted curve on a screen is sampled: the program picks x values across the window, evaluates the rule at each, and joins consecutive results with straight segments. The samples are exact and the segments between them are a guess, so a feature narrower than the gap between samples can be missed entirely and the drawing gives no sign that anything is missing. like that is fine when the rule does nothing between the samples that the samples did not see coming. When it does, the polyline is not a slightly rough version of the truth. It is a different shape, drawn with complete confidence.

Reveal the true curve and compare the two. The gap between them is not measurement error, and nothing was rounded. Every dot your polyline passes through is exactly right, because the rule really does take those values at those inputs. All of the wrongness lives in the segments between the dots, which are the part nobody computed.

The count going up is not the same as the error going down. On the wave, take the count from 7 to 13 and every sample still comes back zero, the line stays flat, and the widest gap between your line and the true curve is 1.0000 at both settings. At 9 samples the gap is 1.3559, worse than at 7. It settles only once the samples are closer together than the feature: 0.2105 at 25, and 0.0020 at 301. The other curve fails in a different way. At four samples it returns 0.005, 0.217, 0.009 and 0.002, and the line through them misses the top of a bump by 0.8137.

Raising the count never proves you have enough. It rules out the features too wide to hide between your samples, and a rule can always have something narrower. The only real defence is knowing something about the rule before you plot it: where it turns, where it has no value, how fast it is able to change. That is what the next step is for.

Meeting curves one formula at a time leaves you with a list to memorise and no way to check it. Six shapes cut the list down to something you can hold. Each one is a parent function The plainest member of a family of curves, the one every other member is a moved, stretched or flipped copy of. x², √x and |x| are parent functions; 3(x − 2)² + 1 is not, it is the parent x² with a = 3, h = 2 and k = 1. The point of the name is that a family is not a list to memorise: knowing the parent and what a, h and k do to it gives you every member, and the same three numbers do the same three things to every parent., the plainest member of its family, and the next two steps turn each parent into the entire family it heads.

f(x) = x is the reference. It is straight, it climbs at the same rate everywhere, and the other five are described by how they fail to do that.

f(x) = x² and f(x) = x³ differ in what they do with negative inputs. Squaring a negative gives a positive, so the left half is folded up to match the right: x² is an even function A function with f(−x) = f(x), whose graph is symmetric about the vertical axis. Cosine and secant are even. The name comes from powers: x², x⁴ and x⁶ have this property, and the power series for cosine contains only even powers. and its graph is symmetric about the vertical axis. Cubing a negative gives a negative, so the left half stays below the axis: x³ is an odd function A function with f(−x) = −f(x), whose graph has rotational symmetry about the origin. Sine, cosecant, tangent and cotangent are odd. The name comes from powers: x, x³ and x⁵ have this property, and the power series for sine contains only odd powers. and its graph has rotational symmetry about the origin. x³ never turns around either. It flattens off near the origin and then keeps climbing.

f(x) = √x lives on half the axis only, because the refusal from the previous page is still in force below 0. It starts at the origin and rises without bound, but ever more slowly.

f(x) = 1/x is the odd one out on every count. It has no value at 0, so it arrives in two separate pieces. Close to 0 it runs away upwards on one side and downwards on the other. Far out in either direction it flattens towards 0 without landing on it. Both of the lines it approaches and never reaches are asymptotes A line a curve approaches arbitrarily closely without ever reaching. In the trigonometric graphs every asymptote is vertical and sits where the definition divides by zero: tangent and secant at the zeros of cosine, cotangent and cosecant at the zeros of sine. The general rule is that a reciprocal function has an asymptote exactly where the original function has a zero. An asymptote need not be vertical: 1/x has a vertical one at x = 0, where it has no value, and a horizontal one at y = 0, which it flattens towards as x runs off in either direction and never lands on.: a vertical one at x = 0, and a horizontal one at y = 0.

f(x) = |x| is the absolute value The size of a number with its sign discarded, written |x|. |3| and |−3| are both 3, and |0| is 0. As a function it is defined on every real number and never returns a negative one, and its graph is two straight half lines meeting at a corner at the origin. |x − a| is the distance between x and a on the number line, which is the reading that matters most later., which returns the size of a number with its sign thrown away, so |−3| and |3| are both 3. Its graph is two straight half lines meeting at a corner at the origin. It resembles x² at a glance and is nothing like it: x² is curved everywhere and has no corner anywhere. The corner matters later, because it is the one place where asking how steep a curve is has no single answer.

y = (x − 3)² + 2 looks like a rule you would need a fresh table of values for. It is x² and nothing else, standing somewhere else on the page. That is what this step and the next are for: a family of curves is not a list to learn, it is one curve and instructions for where to put it.

Take k first, because nothing about it surprises anybody. f(x) + k runs the rule, takes the output, and adds k to it. Every point on the curve goes up by k, or down when k is negative, and nothing else changes. The shape survives untouched because the addition happens after the rule has finished its work.

Now h, and before you touch it, commit. Leave k at zero, take h to a positive value, and decide which way the curve is about to go.

It goes right. Nearly everybody expects left, because the formula contains a subtraction and a subtraction feels like a move backwards.

Here is the reason. f(x − h) does not change what the rule does. It changes what the rule is handed. Give it x, and it works out the bracket first, then asks the parent for its value there. So the height of the moved curve at x is the height the parent had at x − h. To make the bracket come out at 0, which is where the interesting part of most parents sits, you have to hand it x = h. Every feature therefore turns up h further along than it was. The curve reaches a given height when the bracket does, and with h subtracted the bracket needs a bigger x to get there.

Everything the parent has moves with it and nothing is left behind. Slide 1/x and its vertical asymptote A line a curve approaches arbitrarily closely without ever reaching. In the trigonometric graphs every asymptote is vertical and sits where the definition divides by zero: tangent and secant at the zeros of cosine, cotangent and cosecant at the zeros of sine. The general rule is that a reciprocal function has an asymptote exactly where the original function has a zero. An asymptote need not be vertical: 1/x has a vertical one at x = 0, where it has no value, and a horizontal one at y = 0, which it flattens towards as x runs off in either direction and never lands on. leaves x = 0 and stands at x = h. Slide √x and the input it starts at is no longer 0 but h, so the whole domain travels with the curve. Slide |x| and the corner sits at the point (h, k). Change the parent in the figure and the two sliders behave identically on a curve that looks nothing like the last one, which is the claim worth taking away from here.

The third slider does something different in kind from the other two. a multiplies the output after the rule has run, so every height is scaled. A point at height 2 goes to 2a. A point at height −1 goes to −a.

One set of points does not move at all. A point at height 0 stays at height 0 whatever a is, because a · 0 = 0. So the curve is pulled away from the horizontal axis and stays pinned to it wherever it met it. That is the difference between a stretch and a slide. A slide moves every point by the same amount. A stretch moves each point by an amount proportional to how far from the axis it already was.

Above 1, a pulls the curve taller. Between 0 and 1 it squashes the curve flatter. Below 0 it flips the curve top to bottom, a reflection in the horizontal axis, and the size of a still sets how tall. At exactly a = 0 every output the rule produced becomes 0 and the curve flattens onto the horizontal axis. Nothing of the shape survives, so that setting is a degenerate case rather than an interesting one.

The order of operations in a · f(x − h) + k is not decoration. Run one input through it left to right: subtract h, apply the rule, multiply by a, add k. The multiplication happens before the addition, so k is the last thing done, and the curve ends up at height k wherever the parent was at height 0. Write a · (f(x − h) + k) instead and the stretch scales k as well, putting the curve at height a · k. The two forms agree when a = 1, and when k = 0, and differ everywhere else.

Count what you have now. Six parents and three sliders is not eighteen curves to learn. It is six shapes and three motions, and the motions are the same three on every shape. Change the parent and every slider does what it did before. That is checkable in the figure in front of you rather than something to take on trust, so check it.

A hose stands on the lawn on a short stand, pointed up at a fixed 50 degrees, and there is a bucket some distance away on the ground. You want to know where the water will land before you open the tap. Nothing in that question mentions a curve, and the answer is one anyway. The water traces a parabola, so it is x² with some a, h and k, and those three numbers are not yours to pick. The hose fixes them.

Split the speed the water leaves the nozzle at into a sideways part, vx, and an upward part, vy. Nothing acts on vx, so horizontal distance is that speed multiplied by time. Gravity works on the upward part at g = 9.80665 metres per second per second, taking it to zero after vy/g seconds, and that moment is the top of the arc. The top sits at the stand's distance plus vx · vy/g, which is h. Its height is the nozzle height plus vy²/(2g), which is k. The sideways speed alone sets how wide the arc is: a = −g/(2 · vx²), negative because the arc opens downwards.

The panel above the drawing does that arithmetic and prints the equation. The beads are drawn a second time by a second route. Each bead is the water's position at one slice of time, worked out from the two speeds and gravity, and that code never reads a, h or k. So the beads sitting on the drawn curve is a check rather than a decoration. If the algebra above it were wrong, the beads would drift off the line and say so.

Move the stand a metre further out and h grows by exactly a metre, while k and a do not move at all. Raise the nozzle and k rises with it, while h and a stay exactly where they were. Only the tap changes a, and a is the only one of the three that changes the arc's shape. Those are the same three numbers as the sliders in the last two steps, arrived at from a hose rather than from a notation exercise.

Nothing new is taught here. Every curve on this page so far was picked by whoever wrote the page, and a shape you were shown is easier to believe than one you found. This removes that.

The box reads numbers, the letter x, addition, subtraction, multiplication, division, powers, brackets, and the named functions listed beside it. It is a small parser and not a computer algebra system. It will not simplify what you type, it will not solve anything, and it will not rearrange. It reads a formula, works it out at many x values, and joins the results, which is the procedure from the first two steps of this page with your formula in it.

Typing errors are ordinary here and the box expects them. The one people hit first is 2x. On paper two things written next to each other are multiplied, and here the multiplication sign is required. An unclosed bracket is the second. Stopping after ^ with nothing behind it is the third. In each case the message says where reading stopped, and any curve already drawn stays where it is, so a half typed rule never blanks the drawing.

A second kind of failure is not a failure. 1/0 and √(−1) are both legal to type and neither has a value, and the same goes for any named function handed an input outside its own domain. That is the rule refusing, which the previous page spent two steps on, and it is reported as an absence rather than as an error. The drawing carries no point at that input because there is no point to carry.

One thing worth typing before you go wherever you like with it. Take one of the six parents, choose your own a, h and k, and write out a · f(x − h) + k in full with the numbers in it. Set the sliders in the stretching figure earlier on this page to the same three numbers. The two drawings should be the same drawing. That is a check on the two steps that moved a curve, and it does not require believing either of them.

Every number on this page is worked out from the controls as you set them, including the ones in the legends, the table and the compute panel. Nothing here was stored as a picture. The parent curve is shared between the sliding step and the stretching one, so the same shape carries across; every other control belongs to its own figure and starts fresh.