Assignment 4 QA

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course Mth 163

9/22/20141:26 PM

004.

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Question: `q001. Note that this assignment has 7 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.

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Your solution:

The values were as follows:

When x =3 f(x)=13

When x = 7 f(x) = 53

When x = -5 f(x) = 29

Making the points on my graph (3, 13), (7, 53) and (-5, 29). These points look like they could be part of a possible parabola the lowest point being (3, 13) and then the next highest is (-5, 29) but it is on the other side of the y axis highest point being the points (7, 53).

confidence rating #$&*:

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2

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Given Solution:

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.

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Self-critique (if necessary):

I did not remember that this would qualify as one of the previous functions just without the b variable making it 0. So I missed finding the information for the parabola although I did accurately speculate that it was a parabola based on the points and according to my graph the line of symmetry is x =0

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Self-critique rating:

3

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Question: `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.

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Your solution:

F(a) = a^2 +4

F(x+2) = (x+ 2)^2 + 4

X^2 + 4x +8

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

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

I struggled to simplify the next 2 equations without numbers.

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

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

confidence rating #$&*:

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1

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Given Solution:

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.

You should have written these expressions out, and the following should probably be represented on your paper in form similar to that given here:

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Self-critique (if necessary):

I failed to simplify the last 2 equations because I did not write them out correctly on paper. I had trouble using the letters instead of numbers. I’m going to review the answers carefully and try to see the process.

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Self-critique rating:

2

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Question: `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.

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Your solution:

F(x1) = 5*(x1) +7

F(x2) = 5*(x2) +7

[ f(x2) - f(x1) ] / ( x2 - x1 ) =

I do not understand how to complete these problems. The concept is very fuzzy.

confidence rating #$&*:

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1

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Given Solution:

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.

Compare what you have written down with the expressions below:

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Self-critique (if necessary):

This concept is confusing to me. Could you help me understand why these steps have been completed this way????

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You have

[ f(x2) - f(x1) ] / ( x2 - x1 ) =

which indicates that it is this expression that is causing you trouble.

The expression

[ f(x2) - f(x1) ] / ( x2 - x1 )

has numerator

[ f(x2) - f(x1) ]

and denominator

( x2 - x1 ) .

Look at the numerator. It consists of f(x2) and f(x1), with f(x1) subtracted from f(x2).

You have correct expressions for f(x1) and f(x2). What do you get if you substitute those expressions into

[ f(x2) - f(x1) ] ?

Does this agree with the numerator in the given solution?

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In the given solution the numerator is

5 * x2 + 7 - ( 5 * x1 + 7)

What do you get when you simplify this expression, and what steps do you use in doing so?

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Self-critique rating:

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Look at the given solution again. How much of it do you now understand, and what do you not understand?

If you don't see everything, you can submit a copy of this problem with your original solution, the given solution, my notes and your best responses to my questions using the Submit Question Form at

http://vhcc2.vhcc.edu/dsmith/forms/question_form.htm

I'll be glad to respond.

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

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Your solution:

-3 = 5x + 7 subtract 7 from each side

-10 = 5x divide by 5

-2 = x

When x = -2 then f(x) = -3

confidence rating #$&*:

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3

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Given Solution:

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.

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Self-critique (if necessary):

OK

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Self-critique rating:

OK

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

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Question: `q005. If f(x) = 3 x + 2 then what are the values of f(3),f(x+3), 3 f(x) and f(x+h) - f(x)?

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Your solution:

When

f(3) = 3 x + 2

f(x)= 11

f(x+3) = 3 (x+3) + 2

=3x + 9 + 2

=3x +11

3f(x) = 3 x + 2

3(3x + 2)

9x + 6

(Unsure if this is correct)

f(x+ h) - f(x) = 3 x + 2

3(x + h) + 2 - (3 x + 2)

3x + 3 h +2 -3x + 2

= 3h + 4

(Unsure if this is correct)

@&

You aren't doing these correctly.

f(x) = 3 x + 2.

So f(3) is what you get when you replace x by 3.

As an example, let's say we wanted to find f(-5). The -5 takes the place of x in the expression f(x), so in the entire definition

f(x) = 3 x + 2

we would replace x by -5 to get

f(-5) = 3 * (-5) + 2.

The rule is that whatever replaces x in the expression f(x) replaces x in the definition of f(x) (that definition being in this case f(x) = 3 x + 2).

Using that principle, what is f(7)?

What are the following:

f(32)

f(a)

f(q)

f(q+a)

I do recommend that you submit a copy of this question, your solution, my responses and your answers to my questions, as well as another attempt at the original question.

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confidence rating #$&*:

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1

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Question: `q006. If f(x) = 3 x + 2 then what is the value of f(0)? For what value(s) of x do we have f(x) = 0?

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Your solution:

When

F(0) = 3x + 2

= 2

@&

Good, but you want to write

f(0) = 3 * 0 + 2 = 0 + 2 = 2.

In other words, f(x) = 3 x + 2, but f(0) is not equal to 3 x + 2. f(0) is equal to 3 * 0 + 2.

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When

3x + 2 = 0

3x = -2

X = -2/3

@&

Good.

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confidence rating #$&*:

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2

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Question: `q007. Evaluate the expression

• (f(b) - f(a)) / (b - a)

for the function f(x) = 2 ( x - 3 ) + 5.

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Your solution:

[2(x - 3) +5 - 2(x-3) + 5] / b- a

2x - 6 + 5 - 2x -6 + 5 / b - a

= 10 / b - a

I do not think this is correct. I felt like the variables being different would affect the expression and the response.

@&

You're doing pretty well here, but you're being a little careless with signs of grouping, and you haven't replaced x by a in the expression for f(a), or by b in the expression for f(b).

The entire expression for f(a) must be subtracted from the expression for f(b). This required parentheses aroud the expression for f(a). Your numerator

2(x - 3) +5 - 2(x-3) + 5

does simplify to 10, but in order to represent f(a) = f(b) that expression needs to be written

2 ( a - 3) + 5 - ( 2 ( b - 3) + 5)

(be sure to note the parentheses around the expression for f(b).

Give this another try. You should submit it in the same way outlined for the others.

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confidence rating #$&*:

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0

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Self-critique rating:

@&

You're not quite using the function notation correctly, as you have indicated.

I'll help you clarify these if my notes don't lead you to something you're confident in.

See my notes.

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I do not feel like I did well with this assignment I struggled with it immensely. The concept is fuzzy but I can almost see what I am supposed to do.

@&

You aren't far from getting this.

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Self-critique (if necessary):

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Self-critique rating:

@&

You're not quite using the function notation correctly, as you have indicated.

I'll help you clarify these if my notes don't lead you to something you're confident in.

See my notes.

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