Slope: Determine the slope of a linear function

The given linear function is $y = 1 - 4x$. This can be rearranged to fit the standard format of a linear equation, $y = mx + b$, as $y = -4x + 1$.

In this form, the slope $m$ is simply the coefficient of $x$. Here, $m = -4$.

The slope of the linear function represents the rate of change of the function with respect to $x$, meaning for every unit increase in $x$, the value of $y$ decreases by 4 units.

Therefore, the rate of change of the function is $m = -4$, which is option 4 among the given choices.

Thus, the solution to the problem is $m = -4$.

To determine the characteristic of the function $y = 2 - 3x$, we will evaluate the slope:

• The given function is in the form $y = mx + b$, which indicates a linear equation. Here, $y = 2 - 3x$ can be rearranged as $y = -3x + 2$, showing that $m = -3$.
• The slope $m$ is $-3$.
• In a linear function, the sign of the slope $m$ determines the function's behavior:
• If the slope $m$ is positive ($m > 0$), the function is increasing.
• If the slope $m$ is negative ($m < 0$), the function is decreasing.
• If the slope $m = 0$, the function is constant.
• Since $m = -3$, which is negative, we conclude that the function is decreasing.

Therefore, the function described by $y = 2 - 3x$ is decreasing.

The problem asks to find the rate of change of the linear function $y = x - 4$. This function is in the form of $y = mx + b$, where:

• $m$ is the slope, representing the rate of change of the function.
• $b$ is the y-intercept, but it is not relevant for finding the rate of change.

For the function $y = x - 4$, we can compare it with the standard form $y = mx + b$ to identify:

$m = 1$.

Therefore, the rate of change of the function is determined by the coefficient of $x$, which is 1.

Hence, the rate of change of the function is $m = 1$.

The correct answer is: $m = 1$.

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.