course mth 163 pͼȢSQǤƚôassignment #004
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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.
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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
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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.
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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
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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.
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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
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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.
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RESPONSE --> self critique assessment: 3
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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.
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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
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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.
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RESPONSE --> self critique assessment: 3
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19:23:21 `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?
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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
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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.
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RESPONSE --> self critique assessment: 3
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