Part 2: Functions and Quadratics A function consists of: A set of inputs called the domain; A rule...

Part 2: Functions and Quadratics

A function consists of:A set of inputs called the domain;A rule by which each input determines one and only one output;A set of outputs called the range.

Almost all applications in finance and science use functions. An example would be position as a function of time, written x(t): since an object can’t be at two places at the same time, at any time there is one and only one position for the object.

Functions - definition

Functions can often be expressed in equations.

Example: x(t) = xo + vo*t + ½*g*t2 , which is the equation for the position of an object undergoing constant acceleration. The parameters xo, vo, and g are constants. Note that for every t, there is one and only one value for x, so x is a function of t.

Counter-example for functions: from the above, we can express t in terms of x (the inverse relation): t = {-vo ± SQRT[vo

2 – 4(½*g)(xo-x)]}/2((½*g)Note that there is a possibility of an object being at the same place at more than one time, so t is NOT a function of x.


When you graph a function y(x) on your calculator, you will use y for the output (also called the dependent variable) and x for the input (also called the independent variable). If you graph the equation x(t) = xo + vo*t + ½*g*t2 ,it will look roughly likethe drawing on the rightwith t being the x, andx being the y.

t

x


Note in this graph that we can see that there is only one value of x for each value of t - look at the green vertical line..

Also note in this graph that we can see that there are two values of t for each value of x – look at the blue horizontal line.

t

x


How about our linear equation: y = mx + b. Can we say that y is a function of x, y(x)?

For the inverse linear relation, can we say that x is a function of y, x(y)?


How about our linear equation: y = mx + b.

Can we say that y is a function of x, y(x)?Yes, since for every value of x there is one and only one value for y.

Can we say that x is a function of y, x(y)?Solving our linear equation for x gives: x = (1/m)y – b/m. Yes, x(y) is a function since for every value of y there is one and only one value for x.

Functions - domainThe formal definition of a domain is:

Unless specific information to the contrary is given, the domain of a function, f, includes every real number (input) for which the equation (rule) produces a real number as output.

Normally when we think in terms of math, x and y don’t have any limits since real numbers can extend to infinity. However, there are situations that we need to consider, such as the equation (rule) giving us square roots of negative numbers which are NOT real numbers.

Functions – domain

Example: Consider the function

y(x) = √[(x-4)2 - 25]

Is x=3 in the domain for the function?

What would the domain for this function be?


(example continued)

y(x) = √[(x-4)2 - 25]Is x=3 in the domain? NO, since (3-4)2 – 25

= -24, and the square root of a negative number is not a real number.

For the function to return a real number, (x-4)2 ≥ 25, or |x-4| ≥ 5, or x ≥ 9 and x ≤ -1.

Functions - domain

However, in financial and physical terms, particular relations (including functions) apply only under limited circumstances. This is where the domain part of the function definition comes into play.

One example is the force of gravity: F(r) = GMm/r2. Note that F(r) is not defined when r=0. In physics terms mass takes up space, so two masses can’t have zero distance between them since they both can’t occupy the same space at the same time.

Functions - domain

Example: When you throw an object upward on the earth, gravity pulls it back down. If we ignore the effects of air resistance, we get the following function for x(t): x(t) = xo + vo*t + ½*g*t2 .However, this function only applies while the object is in the air. It does NOT apply to before it is thrown, and it does NOT apply for after it hits the ground. Thus the time for which this equation is valid is limited. Time is the independent variable and so this equation has a limited domain.


The concept of domain is helpful when we define a function that has different behaviors during different intervals. Such a function is called a piecewise-defined function.

Example of a piecewise-defined function:A car accelerates from rest for 5 seconds at a

rate of 4 m/s2, and then continues on at a constant velocity for 20 more seconds. Find the function for position versus time.


For the first part of the problem, we need to use the formula for constant acceleration: x(t) = xo + vo*t + ½*a*t2 where xo=0, vo=0 and a=4 m/s2 . This applies for 0 sec. ≤ t ≤ 5 sec. After this, the car continues on at a constant speed (which is determined from v5 = vo + a*t = 20 m/s), so x(t) = x5 + v5*t, and this applies for 5 sec ≤ t ≤ 20 sec. Note that after 5 seconds, x(5 sec) = x5 = ½*(4 m/s2)*(5 sec)2 = 50 m

Functions – domain(example continued from previous slide)

Therefore our function of position versus time is:

x(t) = (2 m/s2)t2 for 0 sec ≤ t ≤ 5 secx(t) = (50 m) + (20 m/s)(t - 5sec)for 5 sec ≤ t ≤ 20 sec

X (m)

T (sec)5 15

50

150

Functions – rate of change

Sometimes we are interested in more than just a value that a function provides. We are interested in how much that value changes when we change the independent variable. This concept has a name: rate of change.

The definition of an average rate of change for a function is simply:

[y(x2) – y(x1)] / [x2 – x1]

Functions – rate of changeExample: x(t) = (501 m) + (20 m/s)*tWe recognize this as a linear function of position with

respect to time (moving at constant speed).The average speed of the object is its average rate of

change of distance during a specified time interval. The average speed between t=5 sec and t=6 sec in this case is: v5-6 = [x(6 sec) – x(5 sec)] / [(6 sec) – (5 sec)] = [(501 m + 120 m) – (501 m +100 m)] / [1 sec] = 20 m/s .

Note that the constant term, 501 m, does not figure into the rate of change. Also note that in this case the slope of the line, 20 m/s, equals the average rate of change, 20 m/s. Is this a coincidence, or is it true in general?


It is easy to see that for a linear function, the slope is constant, and so the average rate of change will also be a constant, and that constant will be the same as the slope.

Average rate of change = [y(x2) – y(x1)] / (x2 – x1) = Δy/Δx, and

Slope = m = (y2 – y1) / (x2 – x1) = Δy/Δx.

For other functions, though, the slopes are not constant, and so the average rates of change also are not constant.


From a graphical point of view, the average rate of change between x2 and x1:

[y(x2) – y(x1)] / (x2 – x1) = Δy/Δx

is the slope of the straight line that goes through the two points (x2, y2) and (x1, y1). This line has a particular name, called the secant line for those two points.


A useful special case of the average rate of change of a function is called the difference quotient, and it is defined as:

[y(x+h) – y(x)] / [(x+h) – x]Or more simply:

[y(x+h) – y(x)] / hThis will be a tool that will be used in

calculus, but it is also a tool we can use now.


difference quotient: [y(x+h) – y(x)] / hExample: Consider the difference quotient (which

is the average rate of change between a point and a near point) for y(x) = ax2 + bx + c. In this case, we have:

[{a(x+h)2 + b(x+h) + c} – {ax2 + bx + c}] / h =[ax2 + 2axh + ah2 + bx + bh + c – ax2 – bx - c] / h =

2ax + b + ah.As h becomes very small, the term (ah) also becomes

very small, and the average rate of change of this function near x is approximately 2ax + b.


The rate of change for y(x) = ax2 + bx + cnear x is approximately 2ax + b.

Note that the constant, c, does not figure into the change (as we should expect).

Note that the bx term gives a constant rate of change, b.Note that the ax2 term gives a rate of change ≈ 2ax. Note

that this term has a rate of change that approximately depends on the independent variable, x. As x increases, the rate of change increases. This should sound plausible since the x2 term goes up much faster the larger x is.


Example: For an object that is falling due to gravity without any air resistance, we know that the object picks up downward speed with time: v(t) = vo + (-g)*t . But we know that speed is the rate of change of position. The result from the previous slide suggests that the position as a function of time should be a quadratic of the form x(t) = xo + vot + ½(-g)t2 , and this turns out to be correct.

Quadratic Functions

Definition of a quadratic function is:y(x) = ax2 + bx + c where a ≠ 0 .The graph of a quadratic function is called a

parabola.The lowest point on an upward opening parabola, or

the highest point on a downward opening parabola is called its vertex.

The vertical line through the vertex, which is a line of reflective symmetry, is called its axis.

Quadratic Functions

y(x) = ax2 + bx + cTo begin our analysis, we note that the x2

term will dominate for large values of x, and so we see that the constant, a, will tell us which way the parabola opens up. If a is positive, the parabola opens upwards. If a is negative, it opens downward.

Try this on your graphing calculator to verify.

Quadratic Functions

y(x) = ax2 + bx + c

Another thing that a does is to determine how fast the parabola rises (or falls). The larger the absolute value of a is, the quicker the parabola rises (or falls), and the narrower the parabola is.

Now we consider what the other constants do.

Quadratic Functions

As we did when we investigated a circle, we will use the tool of “completing the square” to try to see what the constants b, and c do for the parabola.

y(x) = ax2 + bx + cFirst, let’s take a outside: y = a[x2 + (b/a)x + c/a]

Now let’s complete the square for x:y = a[x2 + (b/a)x + (b/2a)2 – (b/2a)2 + c/a](y – [c - b2/4a]) = a(x+b/2a)2

Quadratic Functions

y(x) = ax2 + bx + c(y – [c - b2/4a]) = a(x+b/2a)2

We can identify the vertex as the point where we have 0 = 0 in the blue form, so xv = -b/2a , yv = [c-(b2/4a)]

Verify this for yourself, by choosing values for a, b, and c, and then graphing this function on your calculator and identifying the vertex on your calculator.

Even and Odd Functions

We talked about symmetries earlier. The linear function did not have a symmetry except in the case of the y intercept being zero. In this case, f(x) = bx, the function is symmetric through the origin: f(-x) = -f(x). Functions with this particular symmetry are called odd functions.

What about the function f(x) = ax2 ?We see that this function has a reflection symmetry

about the vertical axis: f(x) = f(-x). Functions with this particular symmetry are called even functions.

Transformations

By graphing functions, we can see how they look. This brings up the question of how can we make them look a little different.

Can we move the graph up or down?Can we move the graph left or right?Can we “reflect” the graph across the

horizontal or vertical axes?Can we stretch or contract the graph?

Transformations

We’ve considered in depth both the linear and quadratic functions, so we have at least two functions to test our ideas with:

linear: y = mx + b

quadratic: y(x) = ax2 + bx + c , or(y – [c - b2/4a]) = a(x+b/2a)2

First: can we move the graph up or down?

Transformationslinear: y = mx + b

quadratic: y(x) = ax2 + bx + c , or(y – [c - b2/4a]) = a(x+b/2a)2

Can we move the graph up and down?If we add a constant, k, to the right side, we will

make y that much bigger for each value of x. linear: y = mx + b + k

quadratic: y(x) = ax2 + bx + c + k

Transformations

linear: y = mx + b +k

quadratic: y(x) = ax2 + bx + c +k, or(y – [c - b2/4a]) = a(x+b/2a)2 +k

We could also subtract that constant, k, from y wherever we see y:linear: (y – k) = mx + bquadratic: (y(x) – k) = ax2 + bx + c, or (y – [c - b2/4a] - k) = a(x+b/2a)2

Check this by trying this on your calculator.

Transformations

Can we perform the same trick for moving things horizontally by adding or subtracting a constant, j, from x?

Transformations

linear: y = m(x-j) + bquadratic: y(x) = a(x-j)2 + b(x-j) + c, or (y – [c - b2/4a]) = a(x+b/2a-j)2


Transformations

Can we expand or contract how high (or low) the graph goes, or how compact or expanded the graph goes?

Transformations

If we simply multiply the whole right side by a factor, p, we will make y bigger (or smaller) by that factor, f:

linear: y = p(mx + b)

quadratic: y(x) = p(ax2 + bx + c) , or(y – p[c - b2/4a]) = ap(x+b/2a)2


Transformations

Could we do the same thing for compressing or expanding in the horizontal direction: replacing every x with gx? Note that this would make x bigger (or smaller) by a factor of g, so if g>1 then gx>x. This would have the effect of compressing the horizontal rather than expanding it.

Try it on your graphing calculator.

Inverse Functions

When is an inverse relation an inverse function?

Recalling that a function, y(x), has one and only one value of y for each value of x, we can see that any function that has a one-to-one relationship between x and y will also have a one-to-one relationship with y and x, and so the y(x) will have an inverse function x(y).

Inverse Functions

Example: Since the linear function, y(x) = mx+b, has only one value of y for each value of x, and it has only one value of x for each value of y, this is a one-to-one relationship and so the linear function has a inverse function: x = (y-b)/m.

Inverse Functions

Counter-example: A counter-example is the quadratic function: y(x) = ax2. There is only one value of y for each value of x (y is a function of x), but there are two values of x that give the same value for y (x and –x give the same y). Therefore, y(x) = ax2 is not a one-to-one relationship, and so x = ± SQRT(y/a) is not a function.

Computer Homework

You should be able to do the fifth Computer Homework assignment on Quadratic Equations, Vol. 0, #4.

Regular Homework Set #5(continued on the next 2 pages)

1. a) Is y(x) = |x - a| a function?b) Graph it.c) If the relation is a function, is it an even or odd function or neither?

2. What is the domain for the function:y(x) = SQRT[(7-x)2 – 36]

3. Given the function: y(x) = 6x2 – 9, a) determine the inverse relation; b) is the inverse relation a function?c) explain your answer to part b.

Regular Homework Set #5(continued from previous page)

4. a) Graph the function of problem 3 above and make a drawing of it.b) Graph the inverse function of problem 2 if it exists; if not, restrict the relation so as to define a function and plot that function.

5. Water flows into an initially empty tank at the rate of 5 liters per minute for 10 minutes; the flow is suddenly reduced to 2 liters per minute and this continues for another 20 minutes.a) Write the function for the amount of water in the tank versus time.b) Draw a graph of this function.


6. Given the function y(x) = 2x3 – 7x, find an expression for the difference quotient.

7. For the quadratic function: y(x) = -2x2 – 7x + 11, find the coordinates of the vertex.

8. For the parabola with vertex (-1,3) and a=1, express the quadratic function in the form: y(x) = ax2 + bx + c (that is, determine the constants a, b, and c).


9) Is the function: y(x) = 7x3 –9x an even function, odd function, or neither. Explain your answer.

10) Express a quadratic function that is the same as y(x) = -2x2 – 7x + 11 (problem 7) but has a vertex that is moved to the right by 3 and down by 4.

Polynomial Functions

What does y = mx + b look like?How many solutions are there for y=0 ?

What does y = ax2 + bx + c look like?How many solutions are there for y=0 ?

Can we generalize for higher degree polynomials?


Can we convert the function y = ax2 + bx + cinto the form: y = a(x-p)(x-q) ? Yes! When you do, you come up with the quadratic equation to give the values of p and q! This shows that when y=0, there are two solutions: x=p and x=q.

Can we convert the function y = ax3 + bx2 + cx + d into the form y = a(x-p)(x-q)(x-r) ? It looks like we should be able to. This means that there are in general three solutions to this third order equation (for any value of y).


For the function y = ax2 + bx + c , as x gets very big (both positive and negative), how does y behave? Is this consistent with there being two solutions in general for each value of y?

For the function y = ax3 + bx2 + cx + d , as x gets very big (both positive and negative), how does y behave? Is this consistent with there being three solutions in general for each value of y?


For the quadratic and the cubic and higher degree polynomials, we need to be aware of the possibility of having repeated roots. For example, y = ax3 + bx2 + cx + d in the form y = a(x-p)(x-q)(x-r) where p and q are the same, what would this graph look like? What about the case where p = q = r ?

Asymptotic Behavior

What happens when the exponent of x becomes negative, such as y(x) = ax-1 = a/x?

How does this function look, and how does it behave? What happens to y as x approaches zero? What happens to y as x approaches very large values? Does it every cross the x or the y axis?

Does the function y(x) = a/(x-b) behave any differently than y(x) = a/x ?

Asymptotic BehaviorThe behavior of y(x) = a/x shows what is

called asymptotic behavior in both the vertical and horizontal directions. As x gets bigger, y approaches zero (the x axis) – we say that the x axis is the horizontal asymptote. As x approaches zero (the y axis), y gets very big – we say that the y axis is the vertical asymptote for this function.

What about y(x) = a/(x-b) ? Does this function have asymptotes?

Asymptotic Behavior

For y(x) = a/(x-b) we see that again see that as x gets very big, y approaches 0 (approaches its horizontal asymptote), and that as x approaches b, y gets very big, so there is a vertical asymptote at x=b.

Can we move the horizontal asymptote up to a value of c?

Asymptotic Behavior

How about if we simply add c to the function: y(x) = a/(x-b) to get: y(x) = a/(x-b) + c. Does this move the horizontal asymptote up by c? YES!

How about other negative powers of x such as y(x) = a/(x-b)2 + c. How does this behave? Does this have vertical and horizontal asymptotes?

Asymptotic Behavior

Counter-example: What kind of function does NOT have asymptotic behavior? Does our original linear function have this asymptotic behavior (where either y or x approaches a definite value but never quite reaches that value)? NO! Does our quadratic function have that behavior? NO!

Rational Functions

We have seen how functions with positive exponents behave (such as y = ax2 + bx + c ) and how functions with negative exponents behave (such as y(x) = a/(x-b) ). How would a combination behave, such as: y(x) = a[mx+b] / [x-b] ? We call this a rational function (in the same way we called a rational number as one integer divided by another).

Rational Functions

y(x) = a[mx+b] / [x-c]

Does this function exhibit asymptotic behavior in either the vertical or horizontal directions?

Rational Functions

y(x) = a[mx+b] / [x-c]

As x approaches c, the denominator approaches zero while the numerator approaches the value a(mc+b). This indicates that y gets larger and larger as x approaches the definite value of c – so there is a vertical asymptote.

As x gets very large, y approaches the value of a – so there is a horizontal asymptote as well.

Rational Functions

What about the rational function: y(x) = [ax2 + bx + c] / [x-d] ? Does it have asymptotes?

Rational Functions

y(x) = [ax2 + bx + c] / [x-d] As x approaches d, the denominator approaches zero

while the numerator approaches the value (ad2+bd+c). This indicates that y gets larger and larger as x approaches the definite value of d – so there is a vertical asymptote.

As x gets very large, y approaches the value of ax which also is getting very large – so no horizontal asymptote in this case, but it does approach the line y=ax which can be considered an asymptote.

WebCT Test #2

You should be able to do the second WebCT test now.

Part 2: Functions and Quadratics A function consists of: A set of inputs called the domain; A rule...

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