Quadratic Graphs; Function Notation

Precalculus I Class Notes 9/04/98

Quadratic Graphs; Function Notation

Graphs of Quadratics

We find the graph of a specific quadratic function using the quadratic formula and what it tells us about the vertex.

Function Notation

We understand function notation f(x) as meaning that f(#expression#) tells us to substitute #expression# for x in the definition of f(x).

We square the expression (a + b) using the distributive law of multiplication over addition, not by using FOIL, which should be abolished. Thus we learn what we need to know to find (a + b) ^ 3, (a + b) ^ 4, etc., and in general to multiply polynomial expressions without resorting to a mindless mnemonic which can't be generalized to anything whatsoever.

Graphs of Quadratics

We begin with the problem of graphing the depth function y = .00226 t^2 - 4.3 t + 92. We will then wish to interpret the meanings of the t and y coordinates of the vertex.

To graph the function we note that it is a quadratic function, and write the equation for the zeros of the function.

We then solve the equation using the quadratic formula, obtaining zeros at t = 951 +- 930.
This expression tells us that the zeros are at t = 951 + 930 =1881 and t = 951 - 930 = 21.
The expression also tells us that the axis of symmetry is at t = 951, halfway between the zeros.
The two zeros are indicated on the t axis, and the axis of symmetry is shown at t = 951.
Note that the quantities that locate the axis of symmetry are circled in green.

http://youtu.be/JklrRCRCKEk

We proceed to use this information to graph the function.

Using function notation y(t) = .00226 t^2 - 4.3 t + 92, we first note that for the zeros t = 1881 and t = 21, we have y(t) = 0; that is,

y(1881) = 0 and
y(21) = 0 (allowing for rounding).

To locate the vertex on the axis of symmetry we need to find the y coordinate of the vertex.

We thus obtain the y coordinate of the vertex by substituting t = 951 into the y(t) function.
We obtain y(951) = .00226 (951)^2 - 4.3 (951) + 92 = -1953.
We thus see that the vertex lies at the point (951, -1953).

Another important point on the graph of any function is the y intercept.

The y intercept is the point on the y axis through which the graph passes.
Since the y axis occurs at t = 0, we substitute 0 for t to obtain this coordinate: y(0) = .00226* (0)^2 - 4.3 * 0 + 92 = 92.
This point is indicated on the graph below, though it is not drawn to scale (if you compare the location of the coordinate y = 92 to that of y = -1953 you will see that the scale is not consistent; this is because the zero at t = 21 should have been drawn much much closer to the y axis).

If we are to interpret the graph in terms of a depth vs. time situation like the one we have seen in class, we will probably restrict our attention to the descending part of the graph, and to clock times t starting at t = 0.

In this case the depth will be modeled by the red-shaded part of the graph, which descends at a decreasing rate until the flow ceases.
It should be clear that flow ceases at the clock time and depth indicated by the vertex.
Thus the flow ceases a clock time t = 951, and that depth -1953.
- (It should be clear that the depth is not measured from the hole from which water flows; if depth is measured in cm, it appears that the hole is 1953 cm, or around 70 feet, below the point from which depth is measured. Perhaps depth is being measured from the top of a tree.)

http://youtu.be/RwIXTHDn_Mg

Function Notation

The quantity we substitute for t need not always been a specific number. We can substitute any expression for t.

For example, if we write f(aardvark), we have implicitly substituted the word aardvark for t (of course 'aardvark' might be the name of a variable, though it is somewhat hard to imagine a situation in which we would give this name to name a variable).
In any case, once we have committed ourselves to substituting aardvark for t, we have to complete the task.
Substituting 'aardvark' for 't', we get f(aardvark) = 2 (aardvark) ^ 2 - 4 (aardvark) + 7.
It's not clear will we should do with this expression, since it doesn't seem to make much sense, but as we have seen we can easily write the expression according to our substitution rules.

http://youtu.be/t5J7jDgwCOk

Whenever we graph of function y = f(t), we find the zeros and y intercept, which are the points were the graph crosses the t axis and the y axis.

The zeros occur on the t axis, where y = 0.

Thus, when we wish to find the zeros, the equation y = f(t) becomes 0 = f(t), or f(t) = 0.
For example for quadratic function y = f(t) = a t^2 + b t + c, f(t) = 0 gives us the quadratic equation a t^2 + b t + c = 0, and we can easily find the zeros using the quadratic formula.

The y intercept occurs on the y axis, which is characterized by the fact that on the y axis t = 0.

Thus the y intercept is characterized by y = f(0).
For example, for quadratic function we find the y intercept of the at y = f(0) = a(0^2) + b(0) + c = c.
The coordinates of the y intercept are therefore (0, c).

For the present function y = 2 t^2 - 4 t + 7, we have

zeros at the values of t which solve the equation 2 t^2 - 4 t + 7 = 0, and
y intercept at (0, c).

Since the function is the quadratic, we also proceed to find the vertex and the points one unit to the right into the left of the vertex.

For the present function we find the vertex is that t = -b / (2a) = -1
When we move one unit to the right or left of the vertex we move up a = 2 units.
We obtain the y coordinate of the vertex by substituting t = -1 into the function, obtain yVertex = f(-1) = ... = 13.
Thus the vertex is (-1, 13). One unit to the right in left we obtain the points (0, 15) and (-2, 15).

http://youtu.be/ZmYW-tvbIk4

When we do mathematics by hand, it is usually convenient to use single letters for function names and variables, since it takes a long time to write down longer variable names.

However when we do mathematics by a computer algebra system, it is often desirable to use sensible names for variables.
If we have three functions f, g and h, using variables x, t and z, then in order to interpret our work we have to remember what each of these symbols means.
It can be difficult and confusing to keep track of the meanings of a bunch of otherwise meaningless symbols.
If we can use sensible names for the functions in the variables, it can make our work more meaningful.
It should be noted, however, that single symbols for variables can be read and comprehended more quickly than longer names, so that single symbols are often more desirable.

Suppose that we are trying to model depth vs. clock time.

Iinstead of writing the function as, e.g., y(t) = 5 * t^2 + 7, we could write it as depth(cTime) = 5 * cTime ^ 2 + 7.
Then we could write things like depth(3) = 5 * 3^2 + 7 = 52, or just depth(3) = 52.
Or if we wish to refer to the depth at a clock time `dt later than cTime, we first note that this clock time is cTime + `dt.
We can then write depth(cTime + `dt) to stand for the depth at this later time.
We can even write the symbolic expression for this depth: depth(cTime + `dt) = 5 * (cTime + `dt) ^ 2 + 7.

The use of the word 'depth' instead of 'y' to symbolize depth, and to a lesser extent 'cTime' instead of t for clock time, makes the meaning of every expression very clear.

However, for the present model it is not at all difficult for most people to remember that y is depth and t is clock time.

In some cases longer and more meaningful names are desirable, while in others they are not.

http://youtu.be/Hc4mhYFCxVI

When we obtained our expression for f(aardvark ), we didn't know what to do with it. However our current expression for depth(cTime + `dt) can be algebraically simplified as in the following example.

http://youtu.be/Yf8a8C0qlVw

Algebra Note

Note that when we square and expression like (a + b), we get

(a + b) ^ 2 = (a + b) ( a + b).

Of course we have a formula for the square of a binomial, but we're going to use this example to illustrate an important principle of algebra.

You might have learned something called FOIL.
The FOIL technique is of questionable value in the teaching of mathematics.
FOIL is a good way to learn to multiply binomials quickly, but it fails to show you what to do a if you have to square, say, a trinomial.
Application of the distributive law of multiplication over addition, on the other hand, is as simple as FOIL.
Application of the laws, as opposed to a limited technique like FOIL, shows how the half-dozen or so laws that express what we know about how numbers behave can be applied.
Application of the laws isn't limited to multiplication of binomoials. The process easily extends to the multiplication of trinomials and even longer expressions, while FOIL doesn't provide a clue.

To apply the laws we proceed as follows:

(a + b) ( a + b) = a (a + b) + b ( a + b), by the distributive law of multiplication over addition.
a(a + b) = a^2 + a b and b (a + b) = ba + b^2, both by the distributive lawful dictation of addition.
Therefore (a + b) ( a + b) = a^2 + ab + ba + b^2.
Since by the commutative law of multiplication, ab = ba, we get

(a + b) ( a + b) = a^2 + ab + ab + b^2 = a^2 + 2 ab + b^2.

If we wanted to square a + b + c, we could use the same idea.

We would have to write out a few more terms, but other than being somewhat longer the process wouldn't be any more difficult, and it would make sense in terms of a few basic laws.

On the other hand, FOIL leaves you clueless with the problem of squaring the trinomial.

Your instructor therefore is of the opinion that the use of FOIL is a pedagocial error. It will be noted that contrary opinions are held by other intelligent individuals.

http://youtu.be/hemCqJrPxS8