 ## MATH1111 Quizzes

Linear Functions Quiz
Web resources available Questions

This quiz tests the work covered in the lecture on linear functions and corresponds to Section 12.4 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There is a great applet at http://www.univie.ac.at/future.media/moe/galerie/geom2/geom2.html which gets you to work out the equations of 5 planes. Make sure you rotate the planes so that you can see where they cut the axes. You need to do some calculations to find the equations.

There are more web quizzes at Wiley, select Sections 4 and 5. This quiz has 14 questions on both this topic and the next.

Consider the linear function $z=3y-2x-1$. Which of the following statements are true? (Zero or more options can be correct)
 a) (3,-2,-1) is a point on the plane. b) The slope of the plane in the $x$ direction is 3. c) The slope of the plane in the $y$ direction is 3 d) The slope of the plane in the $x$ direction is -2. e) The slope of the plane in the $y$ direction is -2.

There is at least one mistake.
For example, choice (a) should be False.
Substitute $x=3$ and $y=-2$ into the equation and we get $z=-6-6-1=-13$ so (3,-2,-1) is not on the plane.
There is at least one mistake.
For example, choice (b) should be False.
The slope of the plane in the $x$ direction is the coefficient of $x$ when the equation of the plane is written in this form, so the slope is -2.
There is at least one mistake.
For example, choice (c) should be True.
The slope of the plane in the $y$ direction is the coefficient of $y$ when the equation of the plane is written in this form, so the slope is 3.
There is at least one mistake.
For example, choice (d) should be True.
The slope of the plane in the $x$ direction is the coefficient of $x$ when the equation of the plane is written in this form, so the slope is -2.
There is at least one mistake.
For example, choice (e) should be False.
The slope of the plane in the $y$ direction is the coefficient of $y$ when the equation of the plane is written in this form, so the slope is 3.
Correct!
1. False Substitute $x=3$ and $y=-2$ into the equation and we get $z=-6-6-1=-13$ so (3,-2,-1) is not on the plane.
2. False The slope of the plane in the $x$ direction is the coefficient of $x$ when the equation of the plane is written in this form, so the slope is -2.
3. True The slope of the plane in the $y$ direction is the coefficient of $y$ when the equation of the plane is written in this form, so the slope is 3.
4. True The slope of the plane in the $x$ direction is the coefficient of $x$ when the equation of the plane is written in this form, so the slope is -2.
5. False The slope of the plane in the $y$ direction is the coefficient of $y$ when the equation of the plane is written in this form, so the slope is 3.
The following table contains the values of a linear function.
$\begin{array}{ccc}\hfill x\setminus y\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 7\hfill & \hfill ?\hfill \\ \hfill \hfill \end{array}$
Which of the following is the missing value and the equation of the plane, respectively? Exactly one option must be correct)
 a) 4 and $z=2x-3y+6\phantom{\rule{0.3em}{0ex}}.$ b) 4 and $z=3x-2y+2\phantom{\rule{0.3em}{0ex}}.$ c) 5 and $z=2x-3y+7\phantom{\rule{0.3em}{0ex}}.$ d) 5 and $z=3x-2y+3\phantom{\rule{0.3em}{0ex}}.$

Choice (a) is incorrect
Try again, the $y$ values are decreasing by 2 and the $x$ values are increasing by 3.
Choice (b) is incorrect
Try again, you have the correct values for the slopes in the $x$ and $y$ directions but you have not calculated the missing value correctly.
Choice (c) is incorrect
Try again, you have calculated the missing value correctly but you do not have the correct equation of the plane.
Choice (d) is correct!
The $x$ values are increasing by 3 and the $y$ values are decreasing by 2 so the missing number is 5 and the equation is $z=3x-2y+c.$ We find $c$ by substituting one of the points into the equation.
$4=3-2+c⇒c=3$ so the equation is $z=3x-2y+3\phantom{\rule{0.3em}{0ex}}.$ It is always a good idea to check another point satisfies this equation as well. ($x=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2$ and $z=2$ from the table which gives 2= 3-4+3 in the equation, which is true)
We are given that the slope of a plane in the $x$ direction is 2 and the slope in the $y$ direction is -2 and (-1,2,5) is a point on the plane.
Which of the following is the equation of that plane? Exactly one option must be correct)
 a) $z=2x-2y+11$ b) $z=2x-2y+7$ c) $z=2y-2x-1$ d) $z+2x-2y+1=0$

Choice (a) is correct!
$z=2x-2y+c$ and we substitute $x=-1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2$ and $z=5$ into the equation to find $c\phantom{\rule{0.3em}{0ex}}.$
$5=-2-4+c⇒c=11$
We could also solve $z-5=2\left(x+1\right)-2\left(y-2\right)$ and we will get the same result.
Choice (b) is incorrect
Try again, you may have used $x=1$ instead of $x=-1\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Try again, the slope in the $x$ direction is the coefficient of $x$ and the slope in the $y$ direction is the coefficient of $y\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Try again, you have not used the correct form for the equation of the plane.
Consider the table below.
$\begin{array}{cccc}\hfill x\setminus y\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 7\hfill \\ \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 10\hfill \\ \hfill 4\hfill & \hfill 10\hfill & \hfill 12\hfill & \hfill 16\hfill \\ \hfill \hfill \end{array}$
Which one of the following statements is correct. Exactly one option must be correct)
 a) The table of values does not represent a linear function. b) The table of values represents the linear function $z=6x+4y-24\phantom{\rule{0.3em}{0ex}}.$ c) The table of values represents the linear function $z=3x+2y-4\phantom{\rule{0.3em}{0ex}}.$ d) The table of values represents the linear function $z=2x+3y-1\phantom{\rule{0.3em}{0ex}}.$

Choice (a) is incorrect
Try again, it does. Note that the $x$ and $y$ values are not evenly spaced.
Choice (b) is incorrect
Try again, you have not calculated the slopes correctly.
Choice (c) is correct!
You have noted that the $x$ and $y$ values are not evenly spaced and that the slope in the $x$ direction is 3 and the slope in the $y$ direction is 2 so the equation is $z=3x+2y-4\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Try again, look carefully at the order of the variables.