Question

The slope of the function is|
x: 1,2,3,4
y: -12, -15, -18, -21
The y-intercept of the function is

Answer 1

To find the slope of the function represented by the given points (1, -12), (2, -15), (3, -18), and (4, -21), we can use the formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Let's calculate the slope using the first two points:

- Point 1: (1, -12) => \( x_1 = 1 \), \( y_1 = -12 \)
- Point 2: (2, -15) => \( x_2 = 2 \), \( y_2 = -15 \)

Now plug these values into the slope formula:

\[
m = \frac{-15 - (-12)}{2 - 1} = \frac{-15 + 12}{1} = \frac{-3}{1} = -3
\]

We can check the slope between another pair of points (for example, (2, -15) and (3, -18)) to verify that we obtain the same result:

- Point 1: (2, -15) => \( x_1 = 2 \), \( y_1 = -15 \)
- Point 2: (3, -18) => \( x_2 = 3 \), \( y_2 = -18 \)

Using the slope formula again:

\[
m = \frac{-18 - (-15)}{3 - 2} = \frac{-18 + 15}{1} = \frac{-3}{1} = -3
\]

This confirms that the slope is consistent for the points provided, and the slope \( m \) is indeed \(-3\).

Now, to find the y-intercept of the function, we can use the slope-intercept form of a linear equation, which is:

\[
y = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept. We can use one of the points to solve for \( b \). Let's use the point (1, -12):

\[
-12 = -3(1) + b
\]

Simplifying this equation:

\[
-12 = -3 + b
\]
\[
b = -12 + 3
\]
\[
b = -9
\]

Thus, the y-intercept of the function is \(-9\).

In summary:
- The slope of the function is \(-3\).
- The y-intercept of the function is \(-9\).