Slope-Intercept Form — Onlayn dərslik

Why They’re Called Linear Relationships
Now it’s time to see where linear relationships get their name. To start, you need to be familiar with the

xy

-plane}.

The

xy

-plane consists of two lines called axes. The

\textbf{x-axis}

goes from left to right (horizontal). The

\textbf{y-axis}

goes from bottom to top (vertical).

Typically, the point where the two axes cross is called the origin (usually marked with an

O

), and it is the point where both the

x

-value and

y

-value are equal to

0

. When working with graphs of mathematical functions, the

x

- and

y

-axes will comply with this arrangement, but it’s important to note that data sets represented as scatter plots, lines of best fit, and other statistical information are sometimes shown on graphs where the axes do not cross at their

0

values, and you must be alert for this situation, as it comes up often on the test.

There are often additional vertical and horizontal lines drawn on the plane that form a grid; these are evenly spaced lines that mark successive

x

- or

y

-values.

Because the

y

-axis goes straight up through the origin, which has an

x

-value of

0

, all the points on the

y

-axis also have an

x

-value of

0

. To the right of the

y

-axis are positive values of

x

, and to the left of the

y

-axis are negative values of

x

. The other vertical gridlines are usually (but not always!) used to denote the successive whole number values of

x

1, 2, 3

, etc. To find the

x

-value of a point from a graph in that common arrangement, just count how many gridlines there are to the left or right of the

y

-axis.

The

x

-axis goes straight through the origin from left to right, and because the origin has a

y

-value of

0

, all the points on the

x

-axis also have a

y

-value of

0

. Above the

x

-axis are positive values of

y

, and below the

x

-axis are negative values of

y

. The other horizontal gridlines are usually (but not always!) used to denote the successive whole number values of

y

1, 2, 3

, etc. To find the

y

-value of a point from a graph like that, count how many gridlines there are above or below the

x

-axis.

Points on this

xy

-plane will often be provided using the notation

(x, y)

where the first number tells us the

x

-value, and the second number tells us the corresponding

y

-value. For example, the point

(4, 2)

4

lines to the right of the

y

-axis because the

x

-value is

4

, and it’s

2

lines above the

x

-axis because the

y

-value is

2

. Because the order in which the two elements appear is important—the first value is the

x

-value, and the second, the

y

-value—this notation represents what is called an ordered pair. The ordered pair that represents the origin is

(0, 0)

because both the

x

- and

y

-values are

0

.
Points on this

xy

-plane will often be provided using the notation

(x, y)

where the first number tells us the

x

-value, and the second number tells us the corresponding

y

-value. For example, the point

(4, 2)

4

lines to the right of the

y

-axis because the

x

-value is

4

, and it’s

2

lines above the

x

-axis because the

y

-value is

2

. Because the order in which the two elements appear is important—the first value is the

x

-value, and the second, the

y

-value—this notation represents what is called an ordered pair. The ordered pair that represents the origin is

(0, 0)

because both the

x

- and

y

-values are

0

Another way you may be given the coordinates of points is based on function notation. For example, if you are told that

f(4) = 2

, then you know that

(4, 2)

is a point on the graph of the function because the function’s

y

-value is equal to

2

when the

x

-value is equal to

4

. In other words, when plotting a function

f

on the

xy

-plane,

y = f(x)

.

The

x

- and

y

-axes divide the

xy

-plane into four quadrants that are numbered with Roman numerals starting in the upper right hand corner and going counter-clockwise.

Now that we know about how points are represented on the

xy

-plane, let’s relate this idea to a real world situation. We will plot some data points based on a scenario and see how they form a line, and linear relationships, when graphed in the

xy

-plane, produce lines—that’s precisely why these relationships are "linear" (line-ar).

Molly is taking a ride on a cable car to the top of a tall building in a big city. The boarding platform is at ground level, and the building, which is located 500 feet away from the boarding dock, is 1,000 feet tall.

If Molly wanted to get to the top of the building without riding the cable car, she would have to walk horizontally 500 ft along the ground and then ride an elevator vertically 1,000 ft. The cable car, on the other hand, travels a straight line path from the dock to the top of the building that covers both the horizontal distance and vertical distance at the same time.

We can represent the cable on the

xy

-plane and use it to find the equation of the linear relationship between the height of the cable car and its horizontal distance from the boarding dock. If we use

x

to represent Molly’s horizontal distance from the dock and

y

to represent her height above the ground, then when Molly is first starting her ride, we can say that

0

is an

x

-value and

0

is the corresponding

y

-value (she is at the boarding dock). Therefore, if we were to graph this point, we would graph the point

(0, 0)

, which is the origin.

In order to define a specific line on a graph, we need at least two points so we can draw a line through both of them. When Molly is at the top of the building, she is 500 feet away horizontally (the

x

-value is 500) from the boarding dock, and she is 1,000 ft above the boarding dock (the

y

-value is 1000), so another point on the graph is

(500, 1000)

. When we graph this point and draw a line connecting the two points we have labelled, we can see a representation of the vertical distance from the boarding dock as a function of the horizontal distance from the boarding dock.

If the boarding dock were built closer to the base of the building, this would affect how slanted the line graph would be. The cable car would have to cover the same vertical distance in a smaller horizontal space. For example, if the boarding dock were only 250 ft away from the building, the tow line would be twice as steep.

We can write an equation to represent Molly’s height as a function of the distance from the boarding dock based on the new, steeper configuration. Her starting height is 0, but we also need to find the amount that her height changes as she moves away from the boarding dock. When we look at any two points on the line, we can find how much Molly moves upward for every foot she moves forward. Let’s use the points

(0, 0)

and

(250, 1000)

.

We can draw two arrows that show how far away the end point at the top of the building lies from the starting point horizontally (in the

x

-direction) and vertically (in the

y

-direction). How far across we go represents the change in

x

-value; we can use the Greek letter delta (

\Delta

) to mean “change in,” so we write the change in

x

-value as

\Delta x

, pronounced “delta

x

.” Similarly, how far up or down we go (that is, the movement parallel to the

y

-axis) represents the change in

y

-value; we write that change in

y

-value as

\Delta y

, pronounced “delta

y

.”

Between the two points in our picture,

(0, 0)

and

(250, 1000)

, we can draw in two arrows (the over-and-up arrows) showing the change in the

x

- and

y

-values.

The

y

-value goes up by

1,000

, so

\Delta y = 1000

. The

x

-value increases by

250

, so

\Delta x = 250

. The word that we use to describe the steepness of such a graphed line is slope, and we define slope as the amount

y

changes relative to a change in

x

, so slope is calculated simply by dividing

\Delta y

(the change in

y

) by

\Delta x

(the change in

x

). For this line, the slope is therefore

\frac{1000}{250}

4

\textbf{Slope}

Slope is a quantity that shows the “steepness” of a line. It is commonly denoted by the variable

m

, and it is calculated using the following formula, where

\Delta y

is the change in

y

-values and

\Delta x

is the change in

x

-values.

m = \frac{\Delta y}{\Delta x}

Alternatively, the change in

y

-values can be more formally written as

y_2 - y_1

, where

y_1

is the

y

-value of the first point and

y_2

is the

y

-value of the second point. The change in

x

-values can be written as

x_2 - x_1

, where

x_1

is the

x

-value of the first point and

x_2

is the

x

-value of the second point. This equation is referred to as the Slope Formula.

and here

m = \frac{y_2 - y_1}{x_2 - x_1}

While this second form of the equation is not incorrect and is sometimes useful, in general, a quicker and less error-prone method of finding the slope is by traveling from one point to the other by drawing the over-and-up arrows (as in the example above) and determining their lengths to find

\Delta y

and

\Delta x

based on the graph alone. It is very easy to mess up signs and values when substituting the coordinates of two points into the Slope Formula. However, if the points are far apart, it could be very time-consuming to do all the incremental counting required using the traveling method, and errors can creep in, so in these circumstances, the Slope Formula is the better choice.

When given points on a line, you should always start by finding the slope whenever possible. It is very rare that the slope is not important to solving a problem involving lines.

For completeness, we will demonstrate using the Slope Formula to find the same information, though we recommend drawing the over-and-up arrows on the graph to find the slope directly. The first point we will choose is

(0, 0)

, which means that

x_1 = 0

and

y_1 = 0

. The second point is

(250, 1000)

, which means that

x_2 = 250

and

y_2 = 1000

. Plug these values into the slope formula and be careful that you are consistent about which point is

(x_1, y_1)

and which point is

(x_2, y_2)

m = \frac{y_2 - y_1}{x_2 - x_1}

m = \frac{1000 - 0}{250 - 0}

m = \frac{1000}{250}

m = 4

The slope of a particular line will be the same no matter which two points are chosen or which point is

(x_1, y_1)

and which point is

(x_2, y_2)

.} What if, instead of

(0, 0)

and

(250, 1000)

, we chose the points

(250, 1000)

and

(125, 500)

(this is the midpoint of the tow line because it is half of the total

x

distance and half of the total

y

-distance between the boarding dock and the top of the building)? Note also that it doesn’t matter whether we draw the over-and-up arrows above or below the line in question, as long as we make sure to go from left to right (it is not wrong to go from right to left, but you have to be careful to take negative signs into account because

\Delta x

would be negative if we go from right to left).

The

y

-value goes up by

500

, so

\Delta y

500

. The

x

-value goes up by

125

, so

\Delta x

125

. The slope is

\frac{500}{125}

, which still simplifies to

4

.
Slope Intercept Form
The next thing to determine when you are trying to find the equation of a line based on a graph is the points where the line crosses the axes. These crossing points are called intercepts.

Specifically, we are interested in the point where the line crosses the

y

-axis. For any linear graph, the point where the line crosses the

y

-axis is called the

y

-intercept}.

\textbf{y-intercept}

The $y$ -intercept is the point where a line crosses the $y$ -axis. Another way of thinking about the

y

-intercept is that it’s the value of $y$ when the $x$ -value is 0 (the

y

-intercept is the value of

f(0)

). This quantity is often represented by the variable

b

.

In the graph below, the

y

-intercept is 2 because the line passes through the point

(0, 2)

, which is on the

y

-axis.

In real world problems, the

y

-intercept represents a starting or base amount when the

x

-value is 0. For example, if your cell phone plan charges you a fixed fee for the line each month plus an additional amount depending on how much data you use, the

y

-intercept of the function representing this linear relationship will be that base fee for the line; that’s what you’d pay even if you use 0 gigabytes of data in a given month. As another example, in the last chapter, we talked about George’s numerous pens, and the number of pens he started his collection with is 20, so 20 is the

y

-intercept.

When you need to solve for the $y$ -intercept of the graph of an equation or function, substitute 0 for $x$ and solve for $y$ .

\textbf{x-intercept}

The $x$ -intercept is the point where a line crosses the $x$ -axis, and it can also be thought of as the value of $x$ when the $y$ -value is 0.

In the graph below, the

x

-intercept is -2 because the line passes through the point

(-2, 0)

, which is on the

x

-axis.

When you need to solve for the $x$ -intercept of the graph of an equation or function, substitute 0 for $y$ and solve for $x$ .

All we have done is given new names to concepts we were already familiar with. The starting amount is the

y

-intercept, and the amount the values change is the slope. We can use this information to define a general linear equation that can be used even when we don’t have a real world context to work from. It’s called the Slope-Intercept Form of a line, and it’s one of the most useful representations of lines on this test.

\textbf{Slope-Intercept Form}

For a line with slope

m

and

y

-intercept

b

, the Slope-Intercept Form of linear equations is the following:

y = mx + b

Note that any of the letters in the Slope-Intercept Form can change; the letters used above are the ones most commonly used by convention, but sometimes different letters that are more meaningful in the context of a specific problem are used instead for clarity. The arrangement and meaning of the terms are what really matter.

In the case of Molly on the cable car, we can find the equation that represents the linear relationship between her height above the ground and her distance (horizontally) from the boarding dock based on just two points on the line. We used the points

(0, 0)

and

(250, 1000)

to find that the slope of the line is 4. Also, we know the

y

-intercept of the line because the cable passes through the point

(0, 0)

; since the

x

-value of the point is 0, this point is the

y

-intercept of the line. Since we know the slope is 4 and the

y

-intercept is 0, the equation that represents the linear relationship in this example is

y = 4x + 0

, or just

y = 4x

As you can see, it’s possible to encounter linear relationships that do not have positive integer slopes, but finding the slope of these lines is just as easy when you are given a graph on which to find the slope. If you are given a real world situation in which the

y

-variable decreases as the

x

-variable increases, you do not necessarily need to graph the points to find the slope or equation of the line. For example, if the average temperature in a certain country drops by 1

^\circ

C every 10 days during the winter, then in a linear equation representing the average temperature in this country during winter, we can say that the slope must be

\frac{-1}{10}

because the temperature decreases by 1 when the number of days increases by 10.
Vertical and Horizontal Lines

\textbf{Vertical Lines}

Vertical lines are lines where the $x$ -value never changes. They have equations like

x = 5

, which tells us that the

x

-value is always 5 and the

y

-value does not depend on the

x

-value at all.

The slope of these lines is undefined, and looking at the Slope Formula will tell us why. Since the

x

-value does not change,

\Delta x = 0

. When we substitute 0 for

\Delta x

in the Slope Formula, we have a problem because it is impossible to divide by 0.

m = \frac{\Delta y}{0} = \text{undefined}

From this, it should be clear that vertical lines have equations written in the form $x = c$ , where $c$ is a constant value. Note, importantly, that this equation is not in Slope-Intercept form, because that form cannot represent a vertical line. For any point on vertical lines, the

x

-value is fixed, but each point will have a different

y

-value.

We can use the Slope Formula to verify that the slope between any two of these points is undefined. From the line

x = 2

, let’s use

(2, 4)

(x_1, y_1)

and

(2, 2)

(x_2, y_2)

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{2 - 2} = \frac{-2}{0} = \text{undefined}

\textbf{Horizontal Lines}

Horizontal lines are lines where the $y$ -value never changes. They have equations like

y = -2

, which tells us that the

y

-value is always

-2

no matter what the

x

-value is.

The slope of these lines is 0. Since the

y

-value does not change,

\Delta y = 0

. When we substitute 0 for

\Delta y

in the Slope Formula, the numerator is 0, and 0 divided by a number is still 0.

m = \frac{0}{\Delta x}

From this, it should be clear that horizontal lines have equations written in the form $y = c$ , where $c$ is a constant value. Note that, unlike the equation for a vertical line, the horizontal line equation is in Slope-Intercept form, as it is equivalent to

y = 0x + c

We can use the Slope Formula to verify that the slope between any two of these points is equal to 0. From the line

y = 2

, let’s use

(4, 2)

(x_1, y_1)

and

(-2, 2)

(x_2, y_2)

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 2}{4 - (-2)} = \frac{0}{6} = 0

Ev tapşırığı