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22:19:30 `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.

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RESPONSE --> if f(x) = x^2 + 4 then: f(3) = 3^2 + 4 = 13 f(7) = 7^2 + 4 = 53 f(-5) = (-5)^2 + 4 = 29 The f(-5) has thrown me a curve...I will self critique when I see what the graph is SUPPOSED to look like!

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22:24:02 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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RESPONSE --> self critique: My graph looked like a parabola (or more a boomerang) passing through (7,53), (-5, 29) and (3,13), I didn't call this a quadratic function since there was no ""b"" ... now I see how it is a quadratic function.

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22:43:51 `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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RESPONSE --> if f(a) = x^2 = 4: f(a)= a^2 = 4 f (x+2) = (x+2)^2 + 4...x(x+2)+2(x+2)+4...x^2+4x+8 f(x+h)=(x+h)^2+4...x(x+h)+h(x+h)+4...x^2+2xh+h^2+4 f(x-h)=(x-h)^2+4...x(x-h)-h(x-h)+4...x^2-2xh+h^2+4 f(x=h)-f(x)=(x+h)^2-x^2+4...x(x+h)+h(x+h)-x^2+4...x^2+2xh+h^2+4 f[f(x+h)-f(x)]/h=2xh+h^2+4/h

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20:18:26 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.

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RESPONSE -->

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20:33:18 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.

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RESPONSE --> After reviewing the answer listed, I saw the mistake made in my calculations. In the set f(x)=x^2 + 4 f (x+h) - f(x), I had not listed a digit. f(x+h) - f(x) = ((x+h)(x+h) + 4) - x^2 + 4 = x(x+h) = (x+h) + 4 - x^2 - 4 = x^2 + xh + xh + h^2 -x^2 = 2xh + h^2 and since the previous answer was incorrect, the latter was also incorrect [f(x + h) - f(x)]/h = ((x + h)(x + h) + 4) - (x^2 + 4)/h = x^2 + 2xh + h^2/h = 2x + h

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20:51:19 `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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RESPONSE --> I am not clear on the subscripted variables and I didn't find it in the class notes. f(x) = 5x + 7 f(x1) = 5(x1) + 7 f(x2) = f(x2) + 7 [ f(x2) - f(x1) ] / (x2-x1) = 5(x2) + 7 - 5(x1) + 7 / x2 - x1 = 5)x2) - 5(x1) / x2 - x1 = 0

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20:55:05 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.

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RESPONSE --> self critique: I made an elementary mistake. I see that the 5 should have been factored out of the numerator.

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20:58:44 `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?

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RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. f(x) = 5x + 7, what is x if f(x) = -3 f(-2)

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20:59:05 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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RESPONSE --> got it!!!!!!!!! yeah

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Your work on this assignment looks very good. Let me know if you have questions.