Assignment 4

course mth 163

pͼȢSQǤƚôassignment #004

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

004.

Precalculus I

01-29-2007

......!!!!!!!!...................................

18:56:06

`q001. Note that this assignment has 4 questions

If f(x) = x^2 + 4, then find the values of the following: f(3), f(7) and f(-5). Plot the corresponding points on a graph of y = f(x) vs. x. Give a good description of your graph.

......!!!!!!!!...................................

RESPONSE -->

For f(3) = (3)^2 + 4 = 9 + 4 = 13

f(7) = (7)^2 + 4 = 49 + 4 = 53

f(-5) = (-5)^2 + 4 = 25 + 4 = 29

The graph when connected in a smooth curve makes a parabola with (3, 13) being the vertex or if in straight lines a check mark of sorts.

confidence assessment: 2

.................................................

......!!!!!!!!...................................

18:58:38

f(x) = x^2 + 4. To find f(3) we replace x by 3 to obtain

f(3) = 3^2 + 4 = 9 + 4 = 13.

Similarly we have

f(7) = 7^2 + 4 = 49 + 4 = 53 and

f(-5) = (-5)^2 + 9 = 25 + 4 = 29.

Graphing f(x) vs. x we will plot the points (3, 13), (7, 53), (-5, 29). The graph of f(x) vs. x will be a parabola passing through these points, since f(x) is seen to be a quadratic function, with a = 1, b = 0 and c = 4.

The x coordinate of the vertex is seen to be -b/(2 a) = -0/(2*1) = 0. The y coordinate of the vertex will therefore be f(0) = 0 ^ 2 + 4 = 0 + 4 = 4. Moving along the graph one unit to the right or left of the vertex (0,4) we arrive at the points (1,5) and (-1,5) on the way to the three points we just graphed.

......!!!!!!!!...................................

RESPONSE -->

I did not go on to find the actual vertex or the points 1 unit each to the right and left, which I should have. I understand how to find them by using the equation -b/2a Then substituting back into the function to get y.

self critique assessment: 2

.................................................

......!!!!!!!!...................................

19:12:13

`q002. If f(x) = x^2 + 4, then give the symbolic expression for each of the following: f(a), f(x+2), f(x+h), f(x+h)-f(x) and [ f(x+h) - f(x) ] / h. Expand and/or simplify these expressions as appropriate.

......!!!!!!!!...................................

RESPONSE -->

Substituting into the funtions:

f(a) = (a)^2 + 4 = a^2 + 4

f(x+2) = [(x+2)(x+2)] + 4 (using foil)

=[x^2 + 2x + 2x + 4] + 4

= x^2 + 4x + 8

f(x+h) = [(x+h)(x+h)] + 4

=(x^2 + 2xh + h^2) + 4

=x^2 + 2xh + h^2 + 4

f(x+h)-f(x) = [(x+h)(x+h) +4] - (x^2 + 4)

=x^2 + 2xh + h^2 + 4 - x^2 - 4 (Combine like terms--the x^2 and 4 are cancelled out)

=2xh +h^2 OR h(2x+h)

[ f(x+h) - f(x) ] / h = {[(x+h)(x+h) + 4] - (x^2 + 4)}/h

=(x^2 + 2xh + h^2 + 4 - x^2 - 4)/h

= (2xh + h^2)/h

=h(2x+h)/h (the h's cancel out leaving

=2x + h

confidence assessment: 3

.................................................

......!!!!!!!!...................................

19:12:56

If f(x) = x^2 + 4, then the expression f(a) is obtained by replacing x with a:

f(a) = a^2 + 4.

Similarly to find f(x+2) we replace x with x + 2:

f(x+2) = (x + 2)^2 + 4, which we might expand to get (x^2 + 4 x + 4) + 4 or x^2 + 4 x + 8.

To find f(x+h) we replace x with x + h to obtain

f(x+h) = (x + h)^2 + 4 = x^2 + 2 h x + h^2 + 4.

To find f(x+h) - f(x) we use the expressions we found for f(x) and f(x+h):

f(x+h) - f(x) = [ x^2 + 2 h x + h^2 + 4 ] - [ x^2 + 4 ] = x^2 + 2 h x + 4 + h^2 - x^2 - 4 = 2 h x + h^2.

To find [ f(x+h) - f(x) ] / h we can use the expressions we just obtained to see that

[ f(x+h) - f(x) ] / h = [ x^2 + 2 h x + h^2 + 4 - ( x^2 + 4) ] / h = (2 h x + h^2) / h = 2 x + h.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

......!!!!!!!!...................................

19:20:37

`q003. If f(x) = 5x + 7, then give the symbolic expression for each of the following: f(x1), f(x2), [ f(x2) - f(x1) ] / ( x2 - x1 ). Note that x1 and x2 stand for subscripted variables (x with subscript 1 and x with subscript 2), not for x * 1 and x * 2. x1 and x2 are simply names for two different values of x. If you aren't clear on what this means please ask the instructor.

......!!!!!!!!...................................

RESPONSE -->

We substitute the numbers into the funtions to arrive at:

f(x1) = 5(x1) + 7

In the same fashion:

f(x2) = 5(x2) + 7

Then taking the responses we just got substitute them into the new equation:

{[5(x2) +7] - [5(x1) + 7]} / ( x2 - x1 )

[5(x2) + 7 - 5(x1) - 7] / ( x2 - x1 ) (Combine like)

[5(x2) - 5(x1)] / ( x2 - x1 ) (Pull out factors)

5(x2-x1) / ( x2 - x1 ) Divide

= 5

confidence assessment: 3

.................................................

......!!!!!!!!...................................

19:20:47

Replacing x by the specified quantities we obtain the following:

f(x1) = 5 * x1 + 7,

f(x2) = 5 * x2 + 7,

[ f(x2) - f(x1) ] / ( x2 - x1) = [ 5 * x2 + 7 - ( 5 * x1 + 7) ] / ( x2 - x1) = [ 5 x2 + 7 - 5 x1 - 7 ] / (x2 - x1) = (5 x2 - 5 x1) / ( x2 - x1).

We can factor 5 out of the numerator to obtain

5 ( x2 - x1 ) / ( x2 - x1 ) = 5.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

......!!!!!!!!...................................

19:23:21

`q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?

......!!!!!!!!...................................

RESPONSE -->

To find the value of x when f(x) = -3 we can set the first function equal to -3

5x + 7 = -3 (then solve for x)

5x = -10 (divide)

x = -2

confidence assessment: 3

.................................................

......!!!!!!!!...................................

19:23:26

If f(x) is equal to -3 then we right f(x) = -3, which we translate into the equation

5x + 7 = -3.

We easily solve this equation (subtract 7 from both sides then divide both sides by 5) to obtain x = -2.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

Your work looks great. Let me know if you have questions.

end of document