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\title{GRAPHING LINEAR FUNCTIONS }


\begin{document}
\maketitle
\section{Objectives} 
\begin{itemize}
\item To recall how to graph straight lines or linear functions.
\item To learn to find x and y intercepts.
\item To learn to find a line knowing a point and its slope.
\item To recognize equations of vertical or horizontal lines.
\end{itemize}
\begin{itemize}
%---------------------------------------------------
\item \textbf{Recall 1:} (Definition of a graph)

Suppose that $f \colon A \rightarrow B$ is a function from $A$ to $B$.  The graph of $f$ is the set of points in the plane described by $(x,f(x))$, where $x$ is in the domain of $f.$  If we plot $(x,f(x))$  for a sufficient number of values of $x$, we find that the points $(x,f(x))$, plotted in the Cartesian plane, are ordered along a curve.

In this Workout, we shall be looking at graphs of functions that come from linear expressions, that is where the variable $x$ is not raised to a power.  For example, $f(x) = 2x - 6.$

The graph of such an expression will be a set of points $(x,f(x))$ that are ordered along a straight line.  Indeed, if we look at a table of values for $(x,f(x))$, we find

\includegraphics{25-1.png}

If we plot these points on the plane, and connect the points by straight lines, we find that all the points lie on the single straight line illustrated below.

\includegraphics{25-2.png}

\end{itemize}
%---------------------------------------------------
\textbf{Note:} Since graphs of functions are plotted with reference to an $x$-axis and a $y$-axis, it is often useful to label the function $y$.  We say that $y = f(x)$, or that $y$ is a function of $x$.  Notice that the value of $f(x)$ is measured along the vertical $y$ axis.  For the same reason, we may write equations, such as $y = 2x - 6$, when we mean $f(x) = 2x - 6.$
%---------------------------------------------------

\begin{itemize} 
\item \textbf{Recall 2:} (The x and y intercepts of a straight line.)

Notice that any straight line that you can draw in the plane  must intersect at least one of the axes, either the $x$-axis or the $y$-axis.  Therefore, straight lines can be classified into three   categories:
\begin{itemize}
\item Lines that intersect only the $x$-axis.
\item Lines that intersect only the $y$-axis.
\item  Lines that intersect both the $x$-axis and the $y$-axis.
\end{itemize}
Lines of type $1$ cannot be the graph of any function of $y$, since they are be vertical lines.  Such lines are completely determined by where they cross the $x$-axis.  For example if  a vertical line crosses the $x$-axis at $x = 9$, then the $x$ value of its   graph must always be $9$.  In other words, the graph must be   the set of all points $(9,y)$, and it does not matter what $y$ is ($y$ can be any real number.)  Recall that a function must have a unique value of $f(x)$ for each value of $x.$

Lines of type $2$ are horizontal lines.  Therefore they have   the same $y$ value for every $x$ value.  Thus, these lines are graphs of functions that take on the same value for all   values of the independent variable $x$.  Points on these lines have the form $(x, b)$, where $b$ is the value at which the line crosses   the $y$-axis.  For example, a horizontal line of height $5$ is the   set of points of the form $(x,5)$ and is the   graph of the function $f(x) = 5$.  A function for which the   value of the function is the same for every value of $x$ is   called a \emph{constant} function.   The graph of any constant   function is always horizontal and it crosses the $y$-axis at the   value of the constant.

Lines of type $3$ intersect  the $x$-axis exactly once and intersect the $y$-axis exactly once.  The point at which the line intersects the $x$-axis   is called the \emph{$x$-intercept}.  The point at which the line   intersects the $y$ axis is called the \emph{$y$-intercept.}

How do we find the intercepts?

 Set $y=0$ to find the $x$-intercept; set $x=0$ to find the $y$-intercept. 
%---------------------------------------------------
\item \textbf{Recall 3.}(The x- and y-intercepts determine the line)

Two points determine a unique line.  We   can see this geometrically.  With two points, we are able   to draw an unambiguous, well-defined, unique line   connecting those two points and extending indefinitely into   space.  How do we determine a line of type $3$?  The easiest   two points to find are the $x$-intercept and the $y$-intercept.    These points will help us graph the line, if we already know   the function. Later we will learn how to find the function,   if we already know the points.

The $x$-intercept is a point of the form $(a,0)$, where $a$ is some fixed   real number.  The $y$-intercept is a point of the form $(0,b)$, where $b$ is   some fixed real number.  These two points can easily be obtained from   the function by setting $y$ equal to $0$ and $x$ equal to $0$,   respectively.  For example, consider the function is $f(x) = 2x - 6$.   By setting $y = f(x) = 0$, we have $0 = 2x - 6$.  Solving   for $x$ gives $x = 3$.  So the $x$-intercept for the graph of this   function is the point $(3,0)$.  Next, we set $x = 0$ to find the $y$-intercept.    Doing so yields $(0,-6)$.  We therefore have two points on   our graph, $(3,0)$ and $(0,-6)$.  Draw the line connecting these   two points to sketch the graph of $f(x) = 2x - 6$.  Notice that   we get the same straight line that appears in Recall $1$; this time we simply plotted two points instead of $13$. 

%---------------------------------------------------

\item \textbf{Recall 4.} (The slope of a line)

If you take another look at the line illustrated in Recall $1$ (the graph of $f(x) = 2x - 6$) you may notice that the $y$ value increases by $2$ units every time the $x$ value increases by $1$ unit.  All lines of type $3$ have the property that their $y$   value increases or decreases each time the $x$ value increases   by one unit.  The amount that the $y$ value increases or   decreases for each unit of increase of the $x$ value is called   the \emph{slope} of the line.  The line given by the function $f(x) =2x - 6$ has a slope of $2$.  That is, $y$ increases by $2$ units every   time $x$ increases by $1$ unit.  To see this, take the point  $(0,-6)$. By increasing $x$ one unit you come to the point on   the line given by $(1,-4)$.  The difference between $-4$ and $-6$   is $2$ units.  The point $(x,y)$ on the line has moved up $2$ units.

It is no accident that the coefficient of the $x$ value of the   function $f(x) = 2x - 6$ is the same as the slope of the line.    In general the graph of a function 
$$y=ax+b$$ 
will be a straight line of slope $a.$  The form of the function above is called the standard form of a line function. In this form $a$ is the slope and $b$ is the $y$-intercept of the line.

Another way of finding the slope is to pick any two points on the graph, say $(x_0,y_0)$ and $(x_1,y_1)$.  Then the difference between the abscissa values $x_1$ and $x_0$  is the horizontal  change, called the run.  The difference between the  ordinate values $y_1$ and $y_0$ is the vertical change, called the rise.  The slope is then calculated as the ratio
$$
\frac{\textrm{rise}}{\textrm{run}}=\frac{y_1-y_0}{x_1-x_0}
$$
For example, if you picked the $x$-intercept $(3,0)$ and the $y$-intercept $(0,-6)$ as the two points, then
$$
\frac{\textrm{rise}}{\textrm{run}}=\frac{0-(-6)}{3-0}=2,
$$
as expected.
%---------------------------------------------------


\item \textbf{Recall 5.} (The $y$-intercept)

  The $y$-intercept has a special use in determining the function from the graph.  If the $y$-intercept and the slope are known, then the function is also known.  All linear functions may be written as
$$f(x) = ax + b,$$
where $a$ is the slope of the graph of $f(x)$ (as we have seen). What is the significance of the constant term   $b$? Try setting $x$ equal to $0$ and notice that $f(0) = b$.  Therefore, $b$ is the value of $y$ at which the   line crosses the $y$ axis. In other words, $b$ is the $y$ value of the $y$-intercept.  This means that by knowing two quantities--the   slope $a$ and the $y$-intercept $b$--we may write the function that   describes the line.  For example, if a line has a slope of $2$ and a $y$-intercept of   $(0,-6)$, then the line is the graph of the function $f(x) = 2x - 6.$
%---------------------------------------------------
\item \textbf{Recall 6.}(How to find the line, knowing the slope and a point)

  The slope of a line may be computed knowing any two   points $(x_0,y_0)$ and $(x_1,y_1).$  We have seen that slope $m$ is given by the ratio
$$\begin{eqnarray}
m=\frac{y_1-y_0}{x_1-x_0}.
\end{eqnarray}
$$
Suppose we know the slope $m=3$ and a point $(3,4)$ of a function $f(x)$, but want to find the rest of the function.  Plugging what we do know into Equation~$1$ gives
$$
3=\frac{y-4}{x-3},
$$
where $x$ and $y$ are unknown variables. We can rearrange this equation to be 
$$
(y-4)=3(x-3).
$$
 Any $x$ and $y$ values that make this equation true will be on the graph of $f(x)$ (can you see why?).  In the same way, we can rearrange Equation~$1$ to give:
$$\begin{eqnarray}
(y-y_0)=m(x-x_0).
\end{eqnarray}
$$
This is the equation of the line with slope $m$ and passing through the point $(x_0,y_0)$.  We call this the \emph{point-slope form} of a line.

%---------------------------------------------------
\item \textbf{Recall 7.} (How to find the line knowing only two points)

  If we know two points $(x_0,y_0)$ and $(x_1,y_1)$, then we can find  the slope of the line through these two points by
$$
m =\frac{(y_1-y_0)}{(x_1-x_0)}.
$$
  From the last Recall, we know how to find the equation of the line by knowing the slope and one point.  Now we know the slope  and the point $(x_0,y_0)$, and we may proceed as we did in   Recall 6. 

\end{itemize}
%---------------------------------------------------

\section{Exercises}\mbox{}\\
1. Two numbers have a relationship where one minus eight times the other equals 57.  The average of the two numbers is 6.  Find the numbers.\\
 

 
2. In a certain far away land, a duke was given on his twentieth birthday a large number of gold ingots.  When he reached 30, he fell in love with a beautiful princess whose fancy he could not easily attract.  After many attempts, he fell to drink and gambling.  At 31, he gambled and lost 1/6 of his gold ingots, at 32 he lost 1/5 of the remaining ingots.  At age 34 he had only 10 ingots left.  How many gold ingots did the duke receive for his twentieth birthday?\\
 

 
3.  The editor of a large publishing company reviews algebra manuscripts for possible publication.  If she rejects 98.5\% of the manuscripts that she reads per year, how many manuscripts does she review for publication, if she accepts 15 new manuscripts per year?\\
 

 
4. A chemical plant discharges diluted waste that contains 0.04\% $H_2S$ (hydrogen sulfide) into a $15,000$ gallon holding container that contains $10,000$ gallons of liquid containing 0.01\% $H_2S.$  The mixture is released into a stream when the concentration reaches 0.02\% $H_2S$.  How much waste is needed before the valves of the holding container are opened to allow the mixture at 0.02\% to flow into the stream?\\
 

 
5. The English Channel tunnel was dug by an English team and a French team.  The French team drilled toward England at the rate of 130 feet per week, while the English team, having poorer drilling equipment drilled at a rate of 90 feet per week.  If it took two years to complete the drilling, how long is the tunnel?\\
 

 
6. A group of young teenagers assemble computers in a garage.  They decide to go into business by selling three different computers.  One contains a single floppy drive and costs \$740.00.  Another has one floppy drive and one hard drive and costs \$1080.00.  A third contains two floppy drives plus a hard drive and costs \$1200.00.  If the costs are entirely due to the drives, what does a floppy drive cost?  What does a hard drive cost?\\
 

 
7. A health food store offers three mixtures of Colombian coffee.  The first contains 5\% chicory and costs \$4.50 a pound.  The second has 5\% chicory and 5\% cocoa bean and costs \$4.55 a pound.  The third has 5\% cocoa bean and costs \$4.75 per pound.  If the price of the coffee is entirely due to the chicory and cocoa bean, what is the price per pound of the cocoa bean?\\
 

 
8. During the Ming Dynasty in China cricket fighting was so much in vogue in the Imperial Palace that the people were permitted to pay taxes by sending crickets.  Taxes were based on the amount of crops that were grown and brought to market.  A bushel of rice could be sold at the market for 40 yuan.  The taxes were assessed at 10\% of market value; but one good fighting cricket could be offered in place of 55 yuan worth of taxes.  If a peasant sold 500 bushels of rice in the market during the year, how many crickets would need if he wished to pay 22\% of his taxes in good fighting crickets?\\
 

 
9. Suppose that the value of a good fighting cricket is as it was in exercise 8.  Also suppose that a bushel of rice sells for 40 yaun but requires 10\% tax, a bushel of maize for 25 yaun requiring 12\% tax and a bushel of potatoes for 12 yaun with 15\% tax.  If a peasant sells 200 bushels of rice, 100 bushels of maize and 50 bushels of potatoes, how many good fighting crickets will he need to pay 50\% of the taxes in good fighting crickets?  (According to Ming tax laws any fractional value of a cricket is equivalent to the next larger whole number of crickets.)\\
 

 
10. The Emperor's cricket was entered in a cricket match.  Tickets were sold at 20 yaun, 40 yaun and 80 yaun each.  Ch'ang-an collected tickets and noticed that the number of 40 yaun tickets was three times the number of least expensive tickets and 50 less than the number of most expensive tickets.   The total amount collected was 8,000 yaun.  How many of each class of tickets was sold?\\
 
 

 
11 Two 19-th Century fictional characters are to fight a dual over a beautiful woman.  They choose their weapons, stand back-to-back and walk in opposite directions.  The taller of the two walks at a rate of one feet per second faster than the shorter.  At the end of 8 seconds they are 56 feet apart.  Find the speed of each person.\\
 

 
12 A troop of soldiers is marching in a line that is 400 feet long.  A messenger at the end of the line must deliver a message to the leader of the troop in the front of the line.  If the troops are marching at a rate of 8 feet per second and the messenger is running at a rate of 10 feet per second, how long will it take for the messenger to deliver his message?\\
 

 
13 In a foot race between Achilles and a tortoise, the tortoise was given a head start of 400 feet.  If Achilles runs at a rate of 20 feet per second and the tortoise runs at a rate of 2 feet per second, how long will it take for Achilles to pass the tortoise? 

%---------------------------------------------------
\section{Solutions} \mbox{}

1. 17 and -5\\

 
2.  15 gold ingots\\

 
3.   1,000 manuscripts\\

 
4.  15,000 gals. of mix or 5,000 gals of waste ?? \\
 
5.  22,880 feet\\

 
6.  \$120.00 for the floppy drive;\\

   \$340.00 for the hard drive.\\
 
7.  \$0.80 per pound\\

 
8.  8 crickets\\

 
9.  11 crickets\\

10. 7 tickets at 20 yaun, 21 tickets at 40 yaun and 71 tickets at 80 yaun\\
 

11. One walks at a rate of 3 feet/sec, the other at 4 feet/sec.\\
 

12. 200 seconds\\

13.$\frac{400}{18}$ seconds or  approx. 22.23 secs \\

\end{document}
