Linear Functions: The Basics

Linear Functions: The Basics

For linear functions, there is very little that you really need to know and keep track of. While there are many variations of questions that may be asked about a line, most of these questions involve two things: 

  1. the slope of the line and 
  2. its y-intercept. 

To determine  these two aspects of the line, all we need to know are either:

  1. Two points on the line, to which you will apply the definition of slope, and
  2. The “slope-intercept” form of the linear function: $y=mx+b$ where $m$ is the slope and $b$ is the y-value of the y-intercept point $(0, b)$.
1. Linear Slope

The slope of a line is simply defined as the “rise per the run” or “the elevation per the horizontal distance” or “the difference of the y-values (rise or elevation) divided by the difference of the x-values (run or horizontal distance)”

The slope ($m$) is defined for a linear function with two arbitrary points $(x_1, y_1)$ and $(x_2, y_2)$ as 

$$m=\frac{\Delta{y}}{\Delta{x}}=\frac{y_2-y_1}{x_2-x_1}$$

It is important to note that a horizontal line, having no change in $y$, has a slope of $0$ since $\frac{0}{x_2-x_1}=0$ for all $x_1 \ne x_2$. Contrastingly, a perfectly vertical line has no change in $x$ and consequently has an undefined slope since $\frac{y_1-y_2}{0}$ for all $y_1 \ne y_2$.

NOTE: Yes, there are other forms that a linear function may take, such as the “point-slope form” or the “standard form,” these other forms are less useful and will likely need to be transformed into “slope-intercept” form at some juncture in any problem. Thus, to save yourself some hassle, you only need to be intimately familiar with $y=mx+b$

Point-Slope Form: $y-y_1=m(x-x_1)$ where $(x_1,y_1)$ represent a point on the line and $m$ = slope

Standard Linear Form: $Ax+By=C$ where $A$, $B$, and $C$ are some constants.

There are two ways of understanding slope apart from the “change in $y$ over the change in $x$”. 

RATE OF CHANGE

Slope is the rate of change in a function; when this function is modeling a real-world phenomenon, then we can say that the slope of the function is the rate of change in whatever two-variable system we are modeling. This kind of wording appears a lot in math texts. If you read or hear “rate of change,” you can safely assume that we are talking about slope in these instances. With this understanding and consideration of “real-world” modeling, we can also better understand the y-axis as the “starting point” or “initial condition” or “initial value” of the system we are modeling. Consider this example problem:

Ex: Fyshstyx bought a piggy bank in March and made an initial deposit of \$5. Fyshstyx then added \$2 to the piggy bank every month. Write an equation to show the total amount of money in Fyshstyx’ account $x$ months after March (assume the account does not accrue interest).

Since the “rate of change” in Fyshstyx’ piggy bank is \$2 every month, we can say that the constant slope of this equation will be 2. Since we begin measuring the amount in the piggy bank at the initial deposit, which is \$5, the point at 0 months ($x=0$) is (0, 5) – that is, the y-intercept of the function will be 5 ($b=5$).

Thus, the equation will be: $y=2x+5$

TANGENT OF AN ANGLE

Slope is the value of the tangent of the angle made between the function and the x-axis. Remembering back to our lessons in basic geometry, the $tan$ function describes the ratio of the $sin$ to $cos$; furthermore, we should recall that under the unit circle coordinate-plane definition of the $sin$ and $cos$ functions: $cos(\theta)=\Delta{x}$ and $sin(\theta)=\Delta{y}$ for the horizontal and vertical distance between the point on the circle and the center of the circle (often at the origin). Thus,

$$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}=\frac{\Delta{y}}{\Delta{x}}=m$$

We might use this understanding to find the slope of an object assuming its angle of elevation or declination.

2. Linear Functions: A Creature by Many Names

Keep in mind that the name of “linear functions” comes in many forms. In different math texts, in different contexts, and on standardized tests like the SAT/ACT, one may see a variety of names for linear functions. Make note of these names so that you know what is meant when you see them; they all mean essentially the same thing and suggest the same essential requirements:

Common Names of Linear Functions:

“line(s)”

“linear graph(s)”

“function(s) with a constant slope”

“directly proportional relationship(s)”

“first degree polynomial(s)”

Sometimes the most difficult thing in math is merely translating human spoken languages into written mathematical statements!

Lines in English Contexts:

Examples: one might see these following phrases in various contexts, but they all aim to say that we are considering a linear function (or line):

“In the x-y plane there is a function with constant slope and points (0, 1) and (c, d)…”

That means the slope is $\frac{d-1}{c}$ and the y-intercept, $b$, is 1 and so $y=\frac{d-1}{c}x+1$

“The variables x and y are in a directly proportional relationship by a ratio of 2:3.”

That means the slope is 2/3, by the way.

There is a polynomial $p(x)$, which has a first degree polynomial factor $f(x)$, where $f(\frac{2}{3})=0$.”

This means that $(3x-2)$ is a factor of this polynomial, $p(x)$, and that $f(x)=3x-2$ is this “first degree polynomial$, which has a slope of 3 and a y-intercept of (0, -2).

3. Linear Inequalities

There is little additional to know about linear inequalities as compared to simple linear functions. Rather than specify a line of values that are true for the function, an inequality (linear or otherwise) establishes a field of possible values that may or may not include the function itself. There are four cases to consider:

$y\gt f(x)$ : in this case, the given function, $f(x)$, establishes a lower bound to the region of solutions, but this boundary is not a valid solution set, since $y$ is greater than (and not equal to) $f(x)$. Due to this, the graphical representation is often given with $f(x)$ as a dotted line with the region above this line shaded in.

$y\ge f(x)$ : in this case, the given function, $f(x)$, establishes a lower bound to the region of solutions with the function itself as a valid solution set, since $y$ is greater than and equal to $f(x)$. Due to this, the graphical representation is often given with $f(x)$ as a solid line with the region above this line shaded in.

$y\lt f(x)$ : in this case, the given function, $f(x)$, establishes an upper bound to the region of solutions, but this boundary is not a valid solution set, since $y$ is less than (and not equal to) $f(x)$. Due to this, the graphical representation is often given with $f(x)$ as a dotted line with the region below this line shaded in.

$y\le f(x)$ : in this case, the given function, $f(x)$, establishes an upper bound to the region of solutions, with the function itself as a valid solution set, since $y$ is less than and equal to $f(x)$. Due to this, the graphical representation is often given with $f(x)$ as a solid line with the region below this line shaded in.

Simply be aware of how these inequalities are graphically represented and where the solutions to such linear inequalities lie (either in a field exclusively or on the function and in a field).

4. Parallel and Perpendicular Lines

We can consider the relationship of two lines according to their orientations to each other. There are two particularly interesting orientations that we should examine: parallel and perpendicular linear functions.

Parallel Lines

PARALLEL LINES are two lines that never intersect in flat space; they travel in precisely the same orientation without ever overlapping. In terms of linear functions we can talk about parallel lines according to their slope. If two lines are parallel, then they must have the same slope. Furthermore, since they cannot intersect, they must also be offset from each other; consequently, they must also have different y-intercepts so that they are not merely the same line. For example, $y=2x+3$ and $y=2x-1$ are parallel lines with slope 2 and y-intercepts of (0, 3) and (0, -1) respectively. 

Perpendicular Lines

PERPENDICULAR LINES are lines that intersect at precisely a right angle (90 degrees or $\frac{\pi}{2}$ radians). These are interesting in terms of linear functions, since if two lines are known to be perpendicular then we can quickly derive the slope of one line if we know the slope of the other. Perpendicular lines have slopes that are opposite reciprocals.

Namely, if the slope of a linear function, $f(x)$, is $m$, then the slope of a linear function $g(x)$ that is perpendicular to $f(x)$ will be $\frac{-1}{m}$. For example, the function $y_1=\frac{3}{5}x+3$ is perpendicular to the function $y_2=\frac{-5}{3}x-4$. Note that the y-intercepts have no effect on the perpendicular nature of the two functions. 

5. Other Linear Ideas to Note (Distance and Midpoint)

Linear Distance Formula (Pythagorean Theorem in an x-y plane)

The Pythagorean Theorem states that for a right triangle with side lengths of $a$ and $b$ and hypotenuse $c$, the square of the length of the hypotenuse $c$ can be described by the sum of the squares of the two sides:

$$a^2+b^2=c^2$$

This formula works just as well in the x-y plane where the distance of $a$ can be thought of as the distance over $x$ between two points, $(x_2-x_1)$, and the distance of $b$ can be thought of as the distance over $y$ between the same two points, $(y_2-y_1)$. Given this, the linear distance, $d$, between two points, $(x_1, y_1)$ and $(x_2, y_2)$ can be derived from the Pythagorean Theorem:

$$d^2=(x_2-x_1)^2+(y_2-y_1)^2$$

To solve for $d$, we take the square root of both sides and get

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Thus, no need to memorize this formula like it is special! It’s just the Pythagorean Theorem in terms of x-y coordinates.

Midpoint of a Line Formula (average of two points)

In very simple terms, the middle of two things is their average: $\frac{a+b}{2}$. We can extend this idea into the x-y plane and say that the midpoint between any two points is simply their average. We find the average of two points by finding the average of the x-values and y-values separately. The point that is made from these averaged values is the mid-point.

For two points in the x-y plane, $(x_1, y_1)$ and $(x_2, y_2)$, the point that lies directly between them (the midpoint) is the average of the x-values and the average of the y-values:

$$(\frac{x_1+x_2}{2} , \frac{y_1+y_2}{2})$$

No need to memorize this formula like it is unique or special! It’s just taking the average of two things for both x-values and y-values.

6. Example Problems

1. What is the slope and y-intercept of the line whose equation is 3x + 21y = 1?

2. Line k in the xy-plane contains points from Quadrants II and III, but no points from Quadrants I or IV. Which of the following must be true?

  • A) The slope of line k is zero.
  • B) The slope of line k is positive.
  • C) The slope of line k is negative.
  • D) The slope of line k is undefined.

3. Derrick filled n glasses of juice from a pitcher. The amount of juice, in ounces, left in the pitcher is given by the expression $128 − 8n$. What is the meaning of $8n$ in this expression?

  • A) Each glass can hold $8n$ ounces of juice.
  • B) The amount of juice in the pitcher was $8n$ ounces.
  • C) A total of $8n$ ounces of juice was poured into the glasses.
  • D) $8n$ ounces of juice is needed to completely fill the pitcher.

4. The line $a$ in the xy-plane passes through (3, 4) and has a slope of $\frac{-5}{6}$. Which point lies on $a$ line $b$ which is perpendicular to line $a$ and also goes through the point (3, 4)?

5.

10. There were 9,600 books in the library at the beginning of the year, and the library receives deliveries of 120 new books each month. Write an expression that represents the total number of books in the library m months from the beginning of the year?

6. Samuel subscribes to a website for music downloads. He does not want to spend more than \$30 on music in a month. If the basic subscription charge is \$10 per month and it costs \$0.80 to download each song, what is the maximum number of songs he can download in a month?

7. The students at Emerson High School measured a tree near their school every year between 2005 and 2015. They noticed that the circumference of the tree can be modeled by the equation $y = 24.5 + 2.4x$, where x represents the number of years since 2005 and y represents the circumference of the tree in centimeters. Which of the following best describes the meaning of the number 2.4 in the equation? 

  • A) The circumference, in centimeters, of the tree in 2005.
  • B) The circumference, in centimeters, of the tree in 2015.
  • C) The total increase in the circumference, in centimeters, between 2005 and 2015.
  • D) The estimated increase in the circumference, in centimeters, of the tree each year.

8. Felipe is filling his parent’s pool, which is 64 inches deep when full. Felipe has been filling the pool for some time, and decides to start a timer to begin tracking his progress. Five minutes after he starts timing, the water is 26 inches from the top of the pool. 10 minutes after he starts timing, the water is 18 inches from the top of the pool. The equation $y = ax + b$ represents the height of the water $y$, in inches, after timing the filling the pool for x minutes. What is the value of $ab$?