Question

Find the rate of change and the initial value of the linear function given as a table of values

X (1, -3)
Y (4, 8)

Answer 1

To find the rate of change (slope) of a linear function given two points, you can use the formula:

\[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}
\]

Given the points (1, -3) and (4, 8):

- Let \( (x_1, y_1) = (1, -3) \)
- Let \( (x_2, y_2) = (4, 8) \)

Now, plug the values into the slope formula:

\[
m = \frac{8 - (-3)}{4 - 1} = \frac{8 + 3}{4 - 1} = \frac{11}{3}
\]

So, the rate of change (slope) is \( \frac{11}{3} \).

Next, to determine the initial value (y-intercept) of the linear function, we can use the slope-intercept form of the equation of a line, which is expressed as:

\[
y = mx + b
\]

where \( b \) is the y-intercept. We can use one of the points we have (let's use \( (1, -3) \)) to solve for \( b \).

Substituting \( m = \frac{11}{3} \) and the coordinates of the point \( (1, -3) \):

\[
-3 = \frac{11}{3}(1) + b
\]

Now, solve for \( b \):

\[
-3 = \frac{11}{3} + b
\]

To isolate \( b \), subtract \( \frac{11}{3} \) from both sides:

\[
b = -3 - \frac{11}{3}
\]

To combine the terms, convert -3 to a fraction with a denominator of 3:

\[
-3 = -\frac{9}{3}
\]

Now:

\[
b = -\frac{9}{3} - \frac{11}{3} = -\frac{20}{3}
\]

Thus, the initial value (y-intercept) of the linear function is \( -\frac{20}{3} \).

In summary:

- Rate of change (slope): \( \frac{11}{3} \)
- Initial value (y-intercept): \( -\frac{20}{3} \)