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# Slope|Definition & Meaning

## Definition

The slope (or **gradient**) is defined as the **ratio** of the difference between two points’ **vertical** and **horizontal** coordinates. Physically, it represents the **steepness** of the line joining the two **points** (how much movement occurs along the **y-axis** for a given movement in the **x-axis** and vice versa). Mathematically:

\[ \text{slope} = m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} \]

**Figure 1** shows two lines with different **slopes**. Line **B** has a greater slope as it is **steeper** than line **A**.

## Line

A line is a **one-dimensional** object which is **infinitely** long and has no width, depth, or curvature. It is made with an infinite number of **points** extending in **opposite** directions. The points lying on the same line are known as **collinear** points.

## Calculating the Slope of a Line

Two points (x_{1}, y_{1}) and (x_{2}, y_{2}) are required to find the slope of a line. The **difference** between the **y-coordinates** gives the **rise**(vertical change), and the difference between the **x-coordinates** gives the **run**(horizontal change).

**Dividing** the rise by run gives the slope “**m**” of the line as:

m = rise/run = (y_{2} – y_{1})/(x_{2} – x_{1})

For example, given two points** (1, 2)** and **(3, 5)** of the line, here:

x_{1} = 1, y_{1} = 2, x_{2} = 3, y_{2} = 5

Calculating the **slope** gives:

m = (5 – 2) / (3 – 1) = 3/2

### Slope in Trigonometry

The **slope** m of a line can also be defined as the **tangent** of the angle **θ** that the line makes with the **x-axis**. Hence,

**m = tan θ**

Where **θ** lies between **-90°** and **90°**. If the **slope** is known, the value of** θ** can be calculated by using the equation:

θ = tan^{-1} (m)

For example, if a line’s **slope** is **10**, the angle **θ** will be:

θ = tan^{-1} (10) = 84.3°

## Infinite and Zero Slope

A **vertical** line has an **infinite** slope, and a **horizontal** line has a **zero** slope. An undefined slope means that with **no change** in x-values, there is a change in y-values. A zero slope means that no change occurs in **y-values** for changing **x-values**.

## Positive and Negative Slope

A **positive** slope means that a **line** moves in the **upward** direction from left to right. The y-values **increase** with increasing x-values.

A **negative** slope of a line means that y-values **decrease** with increasing x-values. The line goes in a **downward** direction from left to right.

**Figure 2** shows lines with zero, infinite, positive, and negative **slopes**.

## Different Forms of Equations of a Line

The equation of a line is known as a **linear** equation having a degree of **one**. The following are **three** different ways to **represent** a linear equation.

### Standard Form

A linear equation’s **standard** form of two variables, **x** and **y**, is:

**Px + Qy = R**

Where “**P**” and “**Q**” are the **coefficients** of the variables **x** and **y**, respectively, and **R** is a constant. For example, the equation **2x – 3y = 9** is in standard form.

### Slope-intercept Form

A line’s** slope-intercept** form is given as

**y = mx + a**

Where “**m**” is the line’s **slope**, and “**a**” is the **y-intercept**.

For example, if a line’s slope is **2** and the y-intercept is **-1**, its **slope-intercept** form will be:

y = 2x – 1

The **y-intercep**t is the value of **y** at the point **(0, a)** where the line **crosses** the **y-axis**. It can be found by placing **x = 0** in the linear equation and solving for **y**.

The **standard** form can be converted into the **slope-intercept** form by rearranging the equation. For example, to **convert** the equation:

x + 2y = 8

into **slope-intercept** form, subtracting **x** on both sides gives:

2y = 8 – x

**Dividing** by **2** into both sides gives:

y = 4 – (1/2)x

Hence, the **slope-intercept** form is:

y = – (1/2)x + 4

#### Graphing a Line Using Slope-Intercept Form

The **slope-intercept** form provides the two essential parameters to **graph** a line, i.e., the slope and the y-intercept. Consider the **equation**:

y = (1/2)x + 4

The **slope** of the line is **1/2**, and its y-intercept is **4**. A slope of **1/2** means a **rise** of **1** and a **run** of **2**. A **positive** slope means that the rise will be in the **upward** direction from left to right.

**Figure 3** shows the **line** obtained by using the **slope-intercept **form.

Starting from the y-intercept **4**, i.e., the point** (0,4)** and moving **1** unit up(**rise**) and 2 units right(**run**), the point reached is **(2, 5)**. A line can be drawn by **joining** the two points (0, 4) and (2, 5) and **extending** the ends infinitely.

### Point-slope Form

A line’s **point-slope** form is given as:

**y – y _{1} = m(x – x_{1})**

Where **x _{1}** and

**y**are the

_{1}**x**and y-coordinates of a

**point**on the line. The point-slope is used to find a line’s

**equation**when a

**point**(x

_{1}, y

_{1}) on the line and its

**slope**are given. It is also used when

**two points**are given with which slope can be calculated.

#### Derivation of Point-slope Form

The point-slope form is **derived** from the **slope-intercept** form. **Figure 4** shows a **line** having any points (x_{1}, y_{1}) and (x, y) on it.

For the point **(x, y)**, the **slope-intercept** form is:

y = mx + a

For the point **(x _{1}, y_{1})**, the slope-intercept form is:

y_{1} = mx_{1} + a

**Subtracting** y_{1} from y gives the **vertical** difference as:

y – y_{1} = (mx + a) – (mx_{1} + a)

y – y_{1} = mx + a – mx_{1} – a

y – y_{1} = mx – mx_{1}

Taking **m** as common gives the **point-slope** form:

**y – y _{1} = m(x – x_{1})**

## Slope of Two Parallel Lines

Two lines are **parallel** if and only if their **slopes** are **equal**. For example, the two lines:

y = 3x + 2 and y = 3x – 5

are **parallel** as both have the **same** slope, i.e., **3** as shown in **figure 5**.

## Slope of Two Perpendicular Lines

Two lines are **perpendicular** to each other if the **product** of their **slopes** is **-1**. For example, the two **lines**:

y = (-1/2)x – 3 and y = 2x + 1

have the slopes m_{1} = **-1/2** and m_{2} = **2** respectively. The **product** of their **slopes** gives:

m_{1}.m_{2} = (-1/2)(2) = -1

Hence, the two **lines** are perpendicular, as shown in **figure 6**.

A line with a **zero** slope is also **perpendicular** to a line with an **infinite** slope.

## Slope of a Function

The **slope** at a point (l, m) on the function’s **curve** can be found by taking its **first derivative** and finding its value by substituting the **point** (l, m). For example, the **function**:

y = x^{3}

has the first **derivative** as:

dy/dx = d(x^{3})/dx = 3x^{2}

To find the **slope** at the point **(2, 3)** on the function, substituting **(2, 3)** in the first **derivative** gives:

dy/dx at P(2, 3) = 3(2)^{2} = 3(4) = 12

Hence, the **slope** at the point (2, 3) is **12**.

### Exponential Function

The **exponential** function **e ^{x}** has the first

**derivative**as:

dy/dx = d(e^{x})/dx = e^{x}

The function f(x) = e^{x} is a **special** function as it is the only function whose **slope** at each point on the curve is equal to the **value** of the **function** at that point. For example, at **x = 1**, the function’s value is:

f(1) = e^{1} = e ≈ 2.178

which is also the **slope** of the curve at the point **(1, e)**, as shown in **figure 7**.

A **function’s** curve has a **varying** slope at each point. A **tangent** line is drawn at the **point** to find the **slope** at that point on the curve. This tangent line’s slope gives the curve’s slope at that point.

## Examples Related to Slope

### Example 1 – Finding the Slope of a Line

Given two points **(0, 0)** and **(5, -2)** on the line, find the **slope**.

### Solution

**Slope** = (y_{2} – y_{1}) / (x_{2} – x_{1}) = (-2 – 0) / (5 – 0) = **-2/5**

### Example 2 – Using the Slope-intercept Form

A line has a slope of** -1** and a y-intercept of** 2**. Find the **equation** of the line and **graph** the line.

### Solution

As the slope and y-intercept are given, the **slope-intercept** form of a line is:

y = mx + a

Here,

m = -1 and a = 2

Putting the **values** gives the equation as:

y = (-1)x + 2

**y = – x + 2**

The slope is -1 with a **rise** of **-1** and a **run** of **1**. To graph the line, starting from point **(0, 2) **and moving **1**(rise) unit **down**(as the slope is negative) and **1**(run) unit **left**. The point reached is **(1, 1).**

Joining the two points **(0, 2)** and** (1, 1)** and extending them in the **opposite** direction is the required **line**, as shown in **figure 8**.

### Example 3 – Using the Point-slope Form

Find the **equation** of a line given the slope as **-2** and a point **(5, 0)** that goes through the line. Also, convert the equation into **standard** form.

### Solution

As the slope and a point are given, the **point-slope** form is:

y – y_{1} = m(x – x_{1})

Here,

(x_{1}, y_{1}) = (5, 0) and m = -2

Putting the **values** in the above equation gives:

y – 0 = -2(x – 5)

**y = -2x + 10**

Adding **2x** on both sides to convert the above equation in **standard** form as:

**2x + y = 10**

*All the images are created using GeoGebra.*