course Mth 163 ¡qíÈ•|·‰ÑéD“ñ\û‹Õ᫳ÜðCÂôassignment #004
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21:30:22 `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 --> f(3)=3^2+4=13 f(7)=7^2+4=53 f(-5)=-5^2+4=29 confidence assessment: 0
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21:33:11 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 was able to find the y value, but I did not understand where to get the coordinated pair. I see now that the x is the value we are substituting for f(x) and the y is the end result so (3,13) (7,53) and (-5,29). I also did not see that b would be 0 since it was in the formula but anything multiplied by 0 will be 0 and can be eliminated from the equation. self critique assessment: 2
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22:02:22 `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 --> f(x)=x^2+4 f(x+2)=(x+2)^2+4 =(x+2)(x+2)+4 =(x^2+2x+2x+4)+4 =x^2+4x+8 f(x+h)=(x+h)^2+4 =(x+h)(x+h)+4 =x^2+hx+hx+h^2+4 =x^2+2hx+h^2+4 f(x+h)-f(x)=[f(x+h)-f(x)]+4 =(x^2+2hx+h^2+4)-(x^2+4 =2hx+h^2+4 [ f(x+h) - f(x) ] / h={[ f(x+h) - f(x) ] / h}^2+4 =[(x^2+2hx+h^2+4)-(x^2+4)]/h =(2hx+h^2+4)/h confidence assessment: 2
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22:03:04 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 --> It looks like I was correct except for the last answer I could have divided further to get the 2x+h self critique assessment: 2
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22:16:05 `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 --> If they stand for different values of x then the expressions would be written as f(x1)=5(x1)+7 f(x2)=5(x2)+7 [ f(x2) - f(x1)]/(x2-x1)=5*[(5(x2)+7)-(5(x1)+7)/(x2-x1 = confidence assessment: 1
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22:16:44 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 --> I was close but the last one could have been simplified more to get a better answer. self critique assessment: 1
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22:19:26 `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3?
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RESPONSE --> -3=5x+7 -3-7=5x+7-7 -10=5x x=-2 confidence assessment: 3
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22:19:36 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 --> ok self critique assessment: 3
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