#$&*
course Mth 163
6/8 around 10:00 am
002.
Note that there are 12 questions in this exercise.
We begin with a short exercise intended to introduce you to the 'basic-points' idea, a very important idea which you will use throughout the course to quickly construct graphs of basic functions and their transformations and combinations. This is a main theme which runs throughout the course.
We then continue with a series of exercises on solving three simultaneous equations in three unknowns, using the process of elimination.
Note on graphs:
Graphs in this course aren't something you get from a graphing calculator. The graphing calculator has its uses and its benefits, but is not needed for most of this course. Graphs are generally constructed using transformations and other techniques, applied to a few simple functions. You will soon understand what this means.
You aren't expected to take the time to do meticulous artwork in this exercise. You will be asked to sketch points on some graphs, and a few curves (parabolas in this case). These graphs are for your own reference, and only need to be neat enough that you can tell what you're seeing. Nobody else needs to see them. Some you might well be able to imagine in your head, with no need to put anything on paper.
You are welcome to use graph paper, but it's probably easier to just sketch a pair of coordinate axes on a piece of paper and mark off an appropriate scale. A set of x and y axes, with each axis running from -8 to 8, will be fine for this exercise. Don't waste time labeling every point on the axes. If you just label the points for -8, -4, 4 and 8 you'll be able to tell what the other coordinates are, and you might not even need to label them (you could just make a larger 'tick mark' at each of these points). You should be able to sketch a usable set of coordinate axes in a minute or less.
It is entirely possible that you'll complete at least some of the first five and understand everything in them by simply following the instructions, then reading the given solutions. If this is the case on one or more of the first five exercises, you can simply put 'OK' for the confidence rating and self-critique rating. By doing so you certify that you have done everything correctly and understand the exercise.
Question `q001: Sketch a set of coordinate axes, with the x axis horizontal and directed to the right, the y axis vertical and directed upwards.
Sketch the point P = (-3, -1) on a set of coordinate axes.
Sketch the point you get if you shift this point -1 units in the horizontal direction. What are the coordinates of your point?
Sketch the point you get if you shift the original point 3 units in the vertical direction. What are the coordinates of your point?
Sketch the point you get if you move the original point 4 times as far from the x axis. What are the coordinates of your point?
If you move the original point 4 times as far from the x axis, then shift the resulting point -1 units in the horizontal direction, and finally shift the point 3 units in the vertical direction, what are the coordinates of the final point?
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Your solution:
If I shift the point P = (-3, -1) -1 units in the horizontal direction I get a point with coordinates (-4, -1).
If I shift the original point 3 units in the vertical direction I get a point with coordinates (-3, 2).
If I move the original point 4 times as far from the x axis I get the point with coordinates (-3, -4).
If I move the original point 4 times as far from the x axis, then shift the resulting point -1 units in the horizontal direction, and finally shift the point 3 units in the vertical direction I get a final point with coordinates (-4, -1).
confidence rating #$&*:
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Self-critique: ok
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Self-critique rating: ok
Question `q002: Starting with the point P = (0, 0):
Sketch the point you get if you shift this point -1 units in the horizontal direction. What are the coordinates of your point?
Sketch the point you get if you shift the original point 3 units in the vertical direction. What are the coordinates of your point?
Sketch the point you get if you move the original point 4 times as far from the x axis. What are the coordinates of your point?
If you move the original point 4 times as far from the x axis, then shift the resulting point -1 units in the horizontal direction, and finally shift the point 3 units in the vertical direction, what are the coordinates of the final point?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your Solution:
The coordinates of my point are (-1, 0).
The coordinates of my point are (0, 3).
The coordinates of my point are (0, 0).
The coordinates of my final point are (-1, 3).
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Self-critique: ok
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Self-critique rating: ok
Question `q003: Plot the points (0, 0), (-1, 1) and (1, 1) on a set of coordinate axes.
Now plot the points you get if you move each of these points 4 times further from the x axis, and put a small circle around each point. What are the coordinates of your points?
Plot the points that result if you shift each of your three circled points -1 units in the x direction. Put a small 'x' through each point. What are the coordinates of your points?
Plot the points that result if you shift each of your three new points (the ones with the x's) 3 units in the y direction. Put a small '+' through each point. What are the coordinates of your points?
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Your Solution:
The coordinates of my points are (0, 0), (-1, 4) and (1, 4).
The coordinates of my points are (-1, 0), (-2, 4) and (0, 4).
The coordinates of my points are (-1, 3), (-2, 7) and (0, 7).
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Self-critique: ok
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Self-critique rating: ok
Question `q004: On the coordinate axes you used in the preceding, sketch the parabola corresponding to the three basic points (0, 0), (-1, 1) and (1, 1).
Then sketch the parabola corresponding to your three circled basic points.
Then sketch the parabola corresponding to three basic points you indicated with 'x's'.
Finally sketch the parabola corresponding to the three basic points you indicated with '+'s'.
Describe how each parabola is related to the one before it.
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Your Solution:
The first parabola will be 4 times further from the x axis than the original parabola. The second parabola is shifted -1 units to the left in the x axis compared to the first parabola. The third parabola has its points 7 times further from the x axis than the points on the original parabola and shifted -1 units to the left in the x axis; and 3 more units in the y axis compared to the second parabola.
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Self-critique: ok
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Self-critique rating: ok
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Question: `q005.
Begin to solve the following system of simultaneous linear equations by first eliminating the variable which is easiest to eliminate. Eliminate the variable from the first and second equations, then from the first and third equations to obtain two equations in the remaining two variables:
2a + 3b + c = 128
60a + 5b + c = 90
200a + 10 b + c = 0.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
-1 (2a + 3b + c = 128) -» expression 1
+ (60a + 5b + c = 90) = -» expression 2
= 58a + 2b = -38 -» expression 4
200a + 10 b + c = 0 -» expression 3
+ -1 ( 2a + 3b + c = 128) = -» expression 1
= 198a + 7b = -128 -» expression 5
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q006. Solve the two equations
58 a + 2 b = -38
198 a + 7 b = -128
which can be obtained from the system in the preceding problem, by eliminating the easiest variable.
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Your solution:
( 58 a + 2 b = -38) (7)
(198 a + 7 b = -128) (-2) =
= (406a + 14b = -266)
+ (-396a - 14b = 256) = (10a)/10 = (-10)/10 =
a= -1
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
Self-critique rating: ok
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Question: `q007. Having obtained a = -1, use either of the equations
58 a + 2 b = -38
198 a + 7 b = -128
to determine the value of b. Check that a = -1 and the value obtained for b are validated by the other equation.
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Your solution:
58 (-1) + 2 b = -38 =
=-58 + 2b = -38 =
2b= 58 -38
(2b)/2= 20/2
b=10
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
Self-critique rating: ok
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Question: `q008. Having obtained a = -1 and b = 10, determine the value of c by substituting these values for a and b into any of the 3 equations in the original system
2a + 3b + c = 128
60a + 5b + c = 90
200a + 10 b + c = 0.
Verify your result by substituting a = -1, b = 10 and the value you obtained for c into another of the original equations.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
2 (-1) + 3 (10) + c = 128
-2 + 30 + c = 128
c = 128 -28
c= 100
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q009. The graph you sketched in a previous assignment contained the given points (1, -2), (3, 5) and (7, 8).
We are going to use simultaneous equations to obtain the equation of that parabola.
• A graph has a parabolic shape if its the equation of the graph is quadratic.
• The equation of a graph is quadratic if it has the form y = a x^2 + b x + c.
• y = a x^2 + b x + c is said to be a quadratic function of x.
To find the precise quadratic function that fits our points, we need only determine the values of a, b and c.
• As we will discover, if we know the coordinates of three points on the graph of a quadratic function, we can use simultaneous equations to find the values of a, b and c.
The first step is to obtain an equation using the first known point.
• What equation do we get if we substitute the x and y values corresponding to the point (1, -2) into the form y = a x^2 + b x + c?
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Your solution:
We get the equation -2 = a(1)^2+b(1)+c
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q010. If a graph of y vs. x contains the points (1, -2), (3, 5) and (7, 8), as in the preceding question, then what two equations do we get if we substitute the x and y values corresponding to the point (3, 5), then the point (7, 8) into the form y = a x^2 + b x + c? (each point will give us one equation)
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
5 = a(3)^2+b(3)+c
8=a(7)^2+b(7)+c
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q011. If a graph of y vs. x contains the points (1, -2), (3, 5) and (7, 8), as was the case in the preceding question, then we obtain three equations with unknowns a, b and c. You have already done this.
Write down the system of equations we got when we substituted the x and y values corresponding to the point (1, -2), (3, 5), and (7, 8), in turn, into the form y = a x^2 + b x + c.
Solve the system to find the values of a, b and c.
• What is the solution of this system?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
-2 = a(1)^2+b(1)+c = a + b +c = -2
5 = a(3)^2+b(3)+c = 9a+3b+c= 5
8=a(7)^2+b(7)+c = 49a + 7b +c =8
(9a+3b+c= 5)
+ -1(a + b +c = -2) = 8a+2b=7
(49a + 7b +c =8) - 1(a + b +c = -2 ) =48 a +6b=10
48 a +6b=10
(-3)(8a+2b=7)= -24 a-6b=-21
(48 a +6b=10) + (-24 a-6b=-21) =
a= -11/24= -0.45833
8(-11/24) +2b=7
= b= 5.33333
-0.45833 + 5.33333 +c = -2
c= - 6.875
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q012. Substitute the values you obtained in the preceding problem for a, b and c into the form y = a x^2 + b x + c, in order to obtain a specific quadratic function.
• What is your function?
• What y values do you get when you substitute x = 1, 3, 5 and 7 into this function?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
My function is y= -0.45833x^2+5.33333x - 6.875
y= -0.45833(1)^2+5.33333(1) - 6.875
y= -2
y= -0.45833(3)^2+5.33333(3) - 6.875
y= 5
y= -0.45833(5)^2+5.33333(5) - 6.875
y=8.33333
y= -0.45833(7)^2+5.33333(7) - 6.875
y=8
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique (if necessary): ok
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Self-critique Rating: ok
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#*&!
&#This looks very good. Let me know if you have any questions. &#