The linear function $y=mx+b$ actually represents a graph of a straight line that has a point of intersection with the vertical $Y$ axis.

$m$ represents the slope.
When $m$ is positive, the slope is positive: the line goes upwards.
When $m$ is negative, the slope is negative: the line goes downwards.
When $m = 0$, the slope is zero: the line is parallel to the $X$ axis.

$b$ represents the point where the line intersects the $Y$ axis.
If $b=0$, then the line will pass through the origin of the coordinates, that is, the point $\left(0,0\right)$

## Examples with solutions for Linear Function y=mx+b

### Exercise #1

Given the linear function:

$y=x-4$

What is the rate of change of the function?

### Step-by-Step Solution

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

In this case, the slope is:

$m=1$

$m=1$

### Exercise #2

Which best describes the function below?

$y=2-3x$

### Step-by-Step Solution

Remember that the rate of change equals the slope.

In this function:

$m=-3$

Therefore, the function is decreasing.

The function is decreasing.

### Exercise #3

Calculate the slope of the line that passes through the points $(4,1),(2,5)$.

### Step-by-Step Solution

Remember the formula for calculating a slope using points:

Now, replace the data in the formula with our own:

$\frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2$

-2

### Exercise #4

What is the slope of a straight line that passes through the points $(0,0),(-8,2)$?

### Step-by-Step Solution

To solve the problem, remember the formula to find the slope using two points

Now, we replace the given points in the calculation:

$\frac{(0-2)}{(0-(-8)}=\frac{-2}{8}=-\frac{1}{4}$

$-\frac{1}{4}$

### Exercise #5

Choose the correct answer for the function.

$y=-x+1$

### Step-by-Step Solution

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.

The graph is parallel to the graph of function

$y=-1-x$

### Exercise #6

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

Negative slope

### Exercise #7

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

Negative slope

### Exercise #8

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

Negative slope

### Exercise #9

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

Positive slope

### Exercise #10

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

Positive slope

### Exercise #11

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

Negative slope

### Exercise #12

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

Negative slope

### Exercise #13

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

Negative slope

### Exercise #14

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

Positive slope

### Exercise #15

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