#$&* course Mth163 Question: `q001. Note that this assignment has 7 questions
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): “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.” Can you explain this a little more to me??? What was the point of these points???
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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: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I thought since they were separated by commas you wanted us to plug in what needed to be plugged in then solve. But I am pretty sure you wanted us to put everything to together now, put all the values together and see what we come up with. I should have replaced the x in f(x) = x^2 + 4 with an a so that f(a)=a^2+4 For the second one replace the x f(x+2)=(x+2)^2+4=x^2+4x+8 (expanded version) To find f(x+h) again replace the x f(x+h)=(x+h)^2+4 expanded x^2+2xh+h^2+4 To find f(x+h)-f(x) you just subtract the pervious problem the original one [x^2+2xh+h^2+4]-[x^2+4] distribute the negative, then you have your product from what is left over after the canceling = 2xh+h^2 To find [ f(x+h) - f(x) ] / h, use the product we just got (so we don’t have to do the pervious problem over again just to get to the first step of this one) Now we have [2xh+h^2]/h the h in 2xh cancels out as well as one of the h’s in h^2 so our answer is = 2x+h Your example really helped. ------------------------------------------------ Self-critique rating:OK
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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: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): [ 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). Sevens cancel out after the negative has been distributed I go plug happy I guess and plugged in the values at the denominator, I’ll watch out for that ------------------------------------------------ Self-critique rating:3 ********************************************* Question: `q004. If f(x) = 5x + 7, then for what value of x is f(x) equal to -3? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&& -3=5x+7 -10=5x x=-2 &&& confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating:OK ********************************************* 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)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&& f(3)=3(3)+2=11 f(x+3)= 3 (x+3)+ 2=3x+11 3 f(x)=3(3x+2)=9x+2