Slope - Examples, Exercises and Solutions

Question Types:
Slope: Using slope for geometric problemsSlope: Determine the slope of a linear functionSlope: Logic and comprehension questionsSlope: Determine whether the slope on the graph is positive or negativeSlope: Calculate the slope from two points

The concept of slope in the function y=mx y=mx expresses the angle between the line and the positive direction of the X X axis.
M M represents the slope of the function – the rate of change of Y Y relative to the rate of change of X X .
When two points on a certain line are known, the slope of the line can be calculated from them. 

If M>0 M>0 is positive - the line rises
If M<0 M<0 is negative - the line falls
If M=0 M=0 the line is parallel to the X X axis. (In a graph like this, where b=0 b=0 the line coincides with the X X axis.)

This calculation is done using the following formula: 

 m=(Y2−Y1)(X2−X1)  m=\frac {(Y2-Y1)}{(X2-X1)}

where the two points (X1,Y1) \left(X1,Y1\right) and (X2,Y2) \left(X2,Y2\right) are on the mentioned line. 

It is important to emphasize that the slope is constant for any line. 

Note:

The greater the slope – the steeper the graph.
The smaller the slope – the more moderate – flatter the graph.
How will you remember this?
Remember that when the slope is equal to 0, the graph is parallel to the X-axis – it is very, very moderate – flat.
Therefore, as it increases, the graph will be steeper.

Detailed explanation

Suggested Topics to Practice in Advance

  1. Function
  2. Linear Function
  3. The Linear Function y=mx+b

Practice Slope

Question 1

Given the linear function:

\( y=x-4 \)

What is the rate of change of the function?

Question 2

Which best describes the function below?

\( y=2-3x \)

Question 3

Choose the correct answer for the function.

\( y=-x+1 \)

Question 4

For the function in front of you, the slope is?

XY

Question 5

For the function in front of you, the slope is?

XY

Examples with solutions for Slope

Exercise #1

Given the linear function:

y=x−4 y=x-4

What is the rate of change of the function?

Video Solution

Step-by-Step Solution

Let's remember that the rate of change equals the slope.

In this case, the slope is:

m=1 m=1

Answer

m=1 m=1

Exercise #2

Which best describes the function below?

y=2−3x y=2-3x

Video Solution

Step-by-Step Solution

Remember that the rate of change equals the slope.

In this function:

m=−3 m=-3

Therefore, the function is decreasing.

Answer

The function is decreasing.

Exercise #3

Choose the correct answer for the function.

y=−x+1 y=-x+1

Video Solution

Step-by-Step Solution

Let's start with option A

In a linear function, to check if the functions are parallel, you must verify if their slope is the same.

y = ax+b

The slope is a

In the original formula:

 y = -x+1

The slope is 1

In option A there is no a at all, which means it equals 1, which means the slope is not the same and the option is incorrect.

 

Option B:

To check if the function passes through the points, we will try to place them in the function:

-1 = -(-2)+1

-1 = 2+1

-1 = 3

The points do not match, and therefore the function does not pass through this point.

 

Option C:

We rearrange the function, in a way that is more convenient:

y = -1-x

y = -x-1

You can see that the slope in the function is the same as we found for the original function (-1), so this is the solution!

 

Option D:

When the slope is negative, the function is decreasing, as the slope is -1, the function is negative and this answer is incorrect.

Answer

The graph is parallel to the graph of function

y=−1−x y=-1-x

Exercise #4

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Exercise #5

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Question 1

For the function in front of you, the slope is?

XY

Question 2

For the function in front of you, the slope is?

XY

Question 3

For the function in front of you, the slope is?

XY

Question 4

For the function in front of you, the slope is?

XY

Question 5

For the function in front of you, the slope is?

XY

Exercise #6

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Exercise #7

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Exercise #8

For the function in front of you, the slope is?

XY

Video Solution

Answer

Positive slope

Exercise #9

For the function in front of you, the slope is?

XY

Video Solution

Answer

Positive slope

Exercise #10

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Question 1

For the function in front of you, the slope is?

XY

Question 2

Given the linear function:

\( y=10-2x \)

What is the rate of change of the function?

Question 3

Given the linear function:

\( y=1-4x \)

What is the rate of change of the function?

Question 4

Given the linear function:

\( y=-6x \)

What is the rate of change of the function?

Question 5

Given the linear function:

\( y=14+5x \)

What is the rate of change of the function?

Exercise #11

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Exercise #12

Given the linear function:

y=10−2x y=10-2x

What is the rate of change of the function?

Video Solution

Answer

m=−2 m=-2

Exercise #13

Given the linear function:

y=1−4x y=1-4x

What is the rate of change of the function?

Video Solution

Answer

m=−4 m=-4

Exercise #14

Given the linear function:

y=−6x y=-6x

What is the rate of change of the function?

Video Solution

Answer

m=−6 m=-6

Exercise #15

Given the linear function:

y=14+5x y=14+5x

What is the rate of change of the function?

Video Solution

Answer

m=5 m=5

Topics learned in later sections

  1. Positive and Negativity of a Linear Function
  2. Finding a Linear Equation
  3. Graphs of Direct Proportionality Functions
  4. Representation of Phenomena Using Linear Functions