MA001: College Algebra

4a. graph points and lines on the Cartesian coordinate plane

• Compare graphing ordered pairs on a coordinate plane with graphing a line on the coordinate plane.

The Cartesian coordinate plane is constructed using two axes that intersect at a $90^{\circ}$ angle. The horizontal axis is typically labeled as the $\boldsymbol{x}-axis$; the vertical axis is typically labeled as the $\boldsymbol{y}-axis$. The point of intersection of the two axes is the origin. Graphing then uses ordered pairs in the form $(x, y)=(a, b)$, with the origin having coordinates $(0,0)$. The first coordinate, or $x$-coordinate, determines the distance left or right from the origin depending on whether $a < 0$ or $a > 0$, respectively. Similarly, the second coordinate, or $y$-coordinate, determines the distance up or down, depending on whether $b > 0$ or $b < 0$, respectively. Thus, the point $(3,5)$ is the point found by moving 3 units to the right of the origin and 5 units up.

To graph a line, find a set of ordered pairs that lie on the equation of the line. Recall from geometry that any two points determine a line. It is often wise to find at least three points that appear to satisfy the equation of the line when graphing to catch any possible errors in identifying ordered pairs on the line.

The Cartesian coordinate plane below shows the two axes, the origin, the point $(3,5)$, and three points that satisfy the equation of the line $y=-\frac{3}{2} x+1$.

Review material in Representations of Linear Functions.

4b. represent a linear function using words, tables, graphs, and function notation

• Define a linear function and express a linear function using function notation.
• Identify characteristics of a linear function when the function is described in words or via a table.

A linear function is a function whose graph is a line. The rule for the linear function only contains the first power of the input variable. In general, a linear function $f$ is described by $\boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{m} \boldsymbol{x} \boldsymbol{+} \boldsymbol{b}$.

One characteristic of a linear function is that it has a constant rate of change of the output variable in relation to the input variable. In function form, this rate of change is denoted by $m$ and represents the slope of the line. The starting value of the linear function is the constant, denoted by $b$ in the function form; this value corresponds to the ordered pair $(0, \, b)$ which is the ordered pair associated with the $\boldsymbol{y}-intercept$ of the graph, or the $y$-value where the graph intersects the $y$-axis. So, a linear function can be identified in a real-world context if there is some starting value and if there is a constant rate of change in the situation. For instance, both situations below can be represented by a linear function.

• An individual opens a savings account with $\$ 100$ and deposits $\$ 50$ each week. If $w$ represents the number of weeks, what function $d(w)$ represents the amount in the account after $w$ weeks if no withdrawals are made? $d(w)=50 w+100$
• A bag of rice contains 50 pounds and a restaurant uses 8 pounds of rice per day. If $x$ represents the number of days, what function $r(x)$ represents the amount of rice left after $x$ days? $r(x)=50-8 x$

If a linear function is represented in a table, the constant rate of change should be visible in the values shown in the table. If the input value changes by a constant amount, then the output value must also change by a constant amount. But be careful! This check can only be applied if the input values change by a constant amount. In the table below, the input values, $x$, increase by 2 from left to right; the output values $f(x)$ increase by 10 from left to right; the output values $g(x)$ do not change by a constant amount from left to right. Hence, $f(x)$ represents a linear function but $g(x)$ does not.

Review material in Representations of Linear Functions.

4c. describe the behavior of a linear function over an interval

• Identify a linear function as increasing, decreasing, or constant.
• Determine the value of the rate of change which leads to an increasing, decreasing, or constant linear function.

As mentioned in objective $4 \mathrm{~b}$, a linear function is one in which there is a constant rate of change. Unless a linear function is restricted to particular values, such as a context in which only non-negative values make sense, both the domain and range of a linear function are represented by the interval $(-\infty, \infty)$. Hence, a linear function is always increasing, always decreasing, or constant.

Recall that a linear function is represented as $f(x)=m x+b$, where $m$ represents the slope or the rate of change. Thus, the linear function is increasing when $m > 0$, decreasing when $m < 0$, and constant when $m=0$. A constant linear function is represented by a horizontal line. On the coordinate grid below, the blue line is increasing, the red line is decreasing, and the black line is constant.

Review material in Representations of Linear Functions.

4d. interpret the slope of a line using words

• Compute the slope of a line.
• Interpret the slope of a line as a change in y for a particular change in x. 
• Contrast a line with zero slope to a line whose slope is undefined.
• Compare the slopes of two lines that are parallel with the slopes of two lines that are perpendicular.

The slope of a line represents the rate of change for the line. For the linear function $f(x)=m x+b$, the slope is represented by $m$ and is found by the following formula:

$m=\frac{\text { change in output }}{\text { change in input }}=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$

When the slope is positive, the line increases from left to right; when the slope is negative, the line decreases from left to right; when the slope is zero, the line is constant, or horizontal. A slope of $\frac{2}{3}$ can be interpreted in several ways, all with equivalent meanings:

• an increase of $\frac{2}{3}$ in the output for every increase of 1 in the input
• a rise of 2 units for every run of 3 units
• up 2 units and to the right 3 units
• down 2 units and to the left 3 units

Horizontal lines have a constant rate of change; that is, there is no change in the output values even when there is a change in the input values. So, the slope of a horizontal line is 0. In contrast, for a vertical line, there is a change in output values even when there is no change in input values. A vertical line is NOT a function, and the slope of a vertical line is undefined.

Two lines are parallel if they are in the same plane and never intersect; parallel lines have the same rate of change, meaning they have the same slope. In contrast, two lines are perpendicular if they intersect at a $90^{\circ}$ angle; the slopes of perpendicular lines are negative reciprocals of each other, meaning that the product of the

slopes of perpendicular lines is $-1$ assuming neither line is vertical. For example, if a line has a slope of $\frac{7}{3}$, a line parallel to it also has a slope of $\frac{7}{3}$; a line perpendicular to it will have a slope of $-\frac{3}{7}$.

Review material in Representations of Linear Functions.

4e. construct the equation of a linear function given two points, a table, or words

• Compare finding the equation of a linear function when given the slope and intercept, two points, or a table of values.
• Explain how characteristics in a context, such as rate of change and starting value, can be used to write a linear function to describe the context.

The slope-intercept form of an equation for a line is $\boldsymbol{y}=\boldsymbol{m x}+\boldsymbol{b}$, where $m$ is the slope and $b$ is the $y$-intercept. If both the slope and the $y$-intercept are known, then these values can be substituted into the slope-intercept form to represent the equation of the line. For instance, if a line has slope 6 and $y$-intercept 2, then the equation of the line is $y=6 x+2$.

If the slope or the $y$-intercept is not given, then the equation of the line can be found by using several steps. Suppose that two points on the line are given. First, find the slope of the line between those two points. Second, substitute the values of the coordinates from one of the ordered pairs into the slope-intercept form to find $b$. Third, use the values of $m$ and $b$ found in the first and second steps to write an equation for the line. For example, suppose $(3,1)$ and $(12,16)$ are two points on a line.

• First, find the slope between these two points: $m=\frac{16-1}{12-3}=\frac{15}{9}=\frac{5}{3}$
• Second: use one of the points as values for $x$ and $y$ together with $m$ to find $b$. Using the ordered pair $(3,1)$

$1=\frac{5}{3} \cdot 3+b$ so $b=-4$

• Third, use $m$ and $b$ to write an equation: $y=\frac{5}{3} x-4$

If the values for a linear function are provided in a table, choose any two sets of input/output values and follow the three steps just listed. If one input value is 0, then the corresponding output value is the $y$-intercept; so, this input/output pair should be chosen when feasible.

In a contextual situation, try to identify the starting value, which represents the $y$-intercept or value for $b$, as well as the constant rate of change, which represents the slope or value for $m$. Consider the following context:

• To join a gym, there is an initial membership fee of $50 and then a monthly fee of $10. Write an equation that gives the total amount paid, $y$, after $x$ months of membership.

The starting value is $b=50$ and the constant change is $m=10$. So, an equation to describe the context is $y=10 x+50$

At times, contextual situations that are linear in nature are best modeled by equations of the form $A x+B y=C$. For instance, consider the following situation:

• At a market, grapes cost $2.49 per pound and apples are \$1.99 per pound. How many pounds $g$ of grapes and $p$ of apples can be purchased for $11.45?

The total cost of $g$ pounds of grapes is $2.49 g$; the cost for $p$ pounds of apples is $1.99 p$. Then, the situation is modeled by the equation $2.49 g+1.99 p=11.45$.

Review material in Representations of Linear Functions.

4f. sketch the graph of a linear function using points, the slope and intercept, equations, and transformations

• Compare graphing a linear function using points, the slope and intercept, or an equation.
• Explain how transformations can be applied to the toolkit linear function $y=x$ to $\mathrm{graph} y=m x+b$.

Several techniques can be used to graph a linear function. Although any technique can be used, one technique may be more efficient than another in a particular situation. If specific points are given, simply connect the line through those points.

If points on the line are not given, then points need to be found. If the slope and $y$-intercept are known, then the linear equation can be graphed by plotting the ordered pair associated with the $y$-intercept and then using the slope as rise over run to find additional points. Given a line with $y$-intercept 2 and slope $\frac{3}{4}$. First graph the point $(0, \, 2)$. Then interpret the slope as a rise of 3 units for a run of 4 units. That is, move 4 units to the right and 3 units up to find the point $(4,5)$; again, move 4 units to the right and 3 units up to obtain the point $\left(8, \, 8\right)$. Now draw the line connecting the points as shown below.

If an equation is given rather than the $y$-intercept and slope, first find points that lie on the line. Although a line can be constructed using just two points, it is useful to find at least three points that lie on the line to catch any errors in evaluating the linear function. It is often efficient to substitute $x=0$ into the equation to find $y$; this value of $y$ is the $y$-intercept. Likewise, it can be useful to substitute $y=0$ into the equation to find $x$; this value of $x$ is the $\boldsymbol{x}-intercept$, namely where the graph crosses the $x$-axis. If substituting either $x=0$ or $y=0$ results in a fraction for the other value, then one might not want to use those values because fractional values are more difficult to plot. Instead, choose a value for one of the variables so that the other variable has an integer value.

For instance, if $3 x+4 y=24$, substituting $x=0$ generates the ordered pair $(0,6)$; substituting $y=0$ gives the ordered pair $(8,0)$; if $x=4$, then $y=3$ to give the ordered pair $(4, \, 3)$. Now draw the line connecting these three points. However, if $5 x+3 y=11$, then substituting $x=0$ gives a fractional value for $y$ and substituting $y=0$ gives a fractional value for $x$. So, try to find values that give integer coordinates. Some such points are (1, 2), $(4, \, -3),(7, \, -8)$; connect the line through these points.

Recall that $f(x)=m x$ is the graph of $f(x)=x$ when stretched vertically by a factor of $m$ if $m > 0$ or compressed vertically by a factor of $m$ if $0 < m < 1$; additionally, if $m < 0$, then the line is reflected vertically across the $x$ axis. Likewise, the graph of $f(x)=x+b$ is the graph of $f(x)=x$ when shifted up if $b > 0$ or down if $b < 0$. When graphing using transformations, any transformations must be completed in a particular order; apply transformations related to $m$ first and then apply transformations related to $b$, similar to multiplying first in the order of operations before adding.

Consider graphing $f(x)=-3 x-4$ using transformations. First, graph $f(x)=x$ (the blue line); two points on this line are $(0, \, 0)$ and $(1, \, 1)$. Second, stretch the line vertically by a factor of 3 (the red line) to obtain the line through the corresponding points $(0, \, 0)$ and $(1, \, 3)$. Third, reflect the line vertically across the $x$-axis (the green line) to obtain the line through the corresponding points $(0, \, 0)$ and $(1, \, -3)$. Finally, shift the line down by 4 units to obtain the graph of $f(x)=-3 x-4$, that is, the black line through the corresponding points $(0,-4)$ and $(1, \, -7)$.

Review material in Representations of Linear Functions.

4g. construct the equation for a linear function given a graph

• Explain how to find an equation for a linear function from a graph.
• Compare the equations for a vertical line with that for a horizontal line.

To find an equation for a line given the graph of the line, use the same techniques as described in objective 4 e. That is, one needs either the slope and $y$-intercept or two points. If the $y$-intercept is an integer and easily identified from the graph, then the value of $b$ in $y=m x+b$ is known. Then, find the slope, perhaps by identifying another ordered pair with integer coordinates and then using the slope formula to find $m$. Once both $m$ and $b$ are known, the equation can be written in the form $y=m x+b$.

If the $y$-intercept is not easily identified, try to find two ordered pairs whose coordinates are integers. Find the slope of the line between those two points. Use the coordinates of one ordered pair in the equation $y=m x+b$ to find $b$ and write an equation for the line.

There are two special cases. A horizontal line represents a linear function. All points on a horizontal line have the same $y$-coordinate. Thus, an equation for a horizontal line is $y=b$. Although a vertical line does not represent a linear function, it is possible to write an equation for a vertical line. All points on a vertical line have the same $x$-coordinate, so an equation for a vertical line is $x=h$.

Review material in Representations of Linear Functions.

4h. make predictions using linear models constructed from words and data

• Explain how to construct a linear model to describe data representing a real-world situation.
• Identify a regression line that fits a set of data.
• Describe how a linear model can be used to predict a future outcome.

Many real-world situations can be modeled by a linear equation. Such situations have a starting value and a constant rate of change of the output variable in relation to the input variable. At times, the data may not lie perfectly along a straight line even though the data appear to be linear in nature. In such cases, draw a line that seems to fit the data with about as many points above the line as below the line; such a line is called a regression line. Then, write an equation for this regression line using the techniques in objectives 4e and 4g. Evaluate the equation for an input value not expressed in the data set to predict the corresponding output variable. In real-world situations, it is important to consider a reasonable domain and range for the set of data.

For instance, consider the set of data in the table below that represent the number of students in a school over several years. The data are then graphed on the coordinate plane and a black line is drawn through the data. For the line that was drawn, the vertical intercept is roughly 425 and the slope of the line is roughly $\frac{75}{6} \approx 12.5$ using the ordered pairs $(0, \, 425)$ and $(6,500)$. So, a possible equation to model the data is $s=12.5 n+425.$ In year 4, the equation predicts a population of 475, which is reasonable given that this population is between the population in year 3 (440 students) and the population in year 5 (485 students). The equation predicts a population of about 675 in year 20. There is no guarantee that the line drawn by hand is the best line to model the data. Finding the best line, called the least-squares regression line is discussed in objective $4 j$.

Review the material in Building Linear Models from Words and Finding the Line of Best Fit.

4i. interpret whether scatter diagrams represent a linear relation

• Explain how the correlation coefficient provides insight into whether a set of data represents a linear relation.

When a set of data is plotted on a coordinate grid, the data may suggest a linear trend, a trend best described by another type of function, or no trend at all. One tool to describe a trend in the data is the correlation coefficient, typically denoted by $\boldsymbol{r}$. Most technology tools that attempt to find an equation to model the data will generate the correlation coefficient.

The correlation coefficient has a value in the interval $-1 \leq r \leq 1$. A positive value of $r$ indicates a relationship with a positive slope; a negative value of $r$ indicates a relationship with a negative slope; a value of $r=0$ indicates no relationship in the data. The closer $r$ is to 1 or $-1$, the stronger the linear relationship; the closer $r$ is to 0, the more scattered the data. Typically, the correlation coefficient is not exactly 1 or $-1$ as such values would indicate a perfect linear relationship with no variability. In most real-world situations that generate data, there is not a perfect relationship.

Review material in Finding the Line of Best Fit.

4j. use a graphing utility to construct a linear regression line given a data set

• Apply a graphing utility to find a least-squares regression line for a set of data.

Most graphing utilities, such as graphing calculators or statistical software, have built-in capabilities for finding a regression line for a set of data. The utilities find the least-squares regression line or the line of best fit, using procedures beyond the scope of this course. Generally, the input values are placed in one statistical list and the output values are placed in a second statistical list. Select the linear regression (linreg) option to find the regression line. Obtain the value of $r$ to determine the strength of the linear relationship.

Caution! Most graphing utilities are able to determine regression equations that model data for a variety of functions, including linear, quadratic, cubic, or logarithmic. Make sure to select linear regression (linreg) and not logarithmic regression (lnreg).

Refer back to the data in objective $4 \mathrm{~h}$. One graphing utility gives the regression equation as $s \approx 19 n+384$ with a value of $0.994$ for the correlation coefficient, indicating a strong positive relationship. Using this equation, we would predict the population for year 4 to be 460 students and for year 20 to be 764 students. These values might be a bit more realistic than those obtained from the regression line that was simply "eye-balled" in objective 4h.

Review material in Finding the Line of Best Fit.

4k. analyze how changes in data affect a regression line

• Explain how a change in the data set might affect the equation for a regression line for the data.

As might be expected, changes in a data set can result in varied changes in a regression line. If the data have a strong linear relationship, then changes in the data may not result in major changes in the regression line; this is particularly true if changes in the data occur in the middle of the data set. In contrast, if the data have a more modest linear relationship, then changes in the data might result in major changes in the equation for the regression line, particularly if the changes in the data occur toward one of the ends of the data set. 

To explore how data changes might affect the regression line, use your graphing utility to recompute the regression line and the related correlation coefficient.

Review material in Finding the Line of Best Fit.

4l. distinguish between interpolation and extrapolation

• Contrast interpolation with a data set and extrapolation with a data set.

One reason to obtain a regression line for a set of data is to enable predictions with its use. That is, the equation is used to obtain output values for input values that are not part of the actual data set. When the prediction is made within the domain and range of the data set, the process is called interpolation. When the prediction is made outside the domain and range of the data set, the process is called extrapolation. For the data set describing the number of students at a school over several years, predicting the student population at year 4 involves interpolation because the given data set has input values from 1 to 15. Predicting the student population for year 20 involves extrapolation because this input value is beyond the data set. As might be expected, extrapolation has a greater tendency to be incorrect than does interpolation. Extrapolation can often lead to unreasonable values because prediction is attempted beyond values where the model makes sense.

Review material in Finding the Line of Best Fit.

Unit 4 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• correlation coefficient, $r$
• extrapolation
• function notation for a linear function
• horizontal axis, $x$-axis
• interpolation
• least-squares regression line
• linear function
• origin
• parallel lines
• perpendicular lines
• regression line
• slope
• slope-intercept form of a linear equation, $y=m x+b$ $x$-intercept
• vertical axis, $y$-axis
• $y$-intercept