To determine the [tex]\( y \)[/tex]-intercept of the function with the greater rate of change, we need to follow these steps:
1. Calculate the slope of the first linear function:
Given the points [tex]\((10, 5)\)[/tex] and [tex]\((-15, -5)\)[/tex], the slope (rate of change) of a line can be calculated using the formula:
[tex]\[
m = \frac{y2 - y1}{x2 - x1}
\][/tex]
Substituting the given points:
[tex]\[
m_1 = \frac{-5 - 5}{-15 - 10} = \frac{-10}{-25} = \frac{2}{5} = 0.4
\][/tex]
So, the slope of the first function is [tex]\( 0.4 \)[/tex].
2. Calculate the [tex]\( y \)[/tex]-intercept of the first linear function:
The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept. Rearranging this equation to solve for [tex]\( b \)[/tex]:
[tex]\[
b = y - mx
\][/tex]
Using one of the points, let's use [tex]\((10, 5)\)[/tex]:
[tex]\[
b_1 = 5 - (0.4 \times 10) = 5 - 4 = 1
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the first function is [tex]\( 1 \)[/tex].
3. Convert the second function into slope-intercept form and find its rate of change and [tex]\( y \)[/tex]-intercept:
The second function is given by the equation [tex]\( 4x - 3y = 6 \)[/tex]. To convert it into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
-3y = -4x + 6
\][/tex]
Dividing every term by [tex]\(-3\)[/tex]:
[tex]\[
y = \frac{4}{3}x - 2
\][/tex]
From this equation, we can see that the slope ([tex]\( m \)[/tex]) of the second function is [tex]\( \frac{4}{3} \)[/tex] and the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is [tex]\(-2\)[/tex].
4. Compare the slopes of both functions:
The slope of the first function ([tex]\( m_1 \)[/tex]) is [tex]\( 0.4 \)[/tex].
The slope of the second function ([tex]\( m_2 \)[/tex]) is [tex]\( \frac{4}{3} \approx 1.333 \)[/tex].
Since [tex]\( 1.333 \)[/tex] (which is [tex]\( \frac{4}{3} \)[/tex]) is greater than [tex]\( 0.4 \)[/tex], the second function has the greater rate of change.
5. Determine the [tex]\( y \)[/tex]-intercept of the function with the greater rate of change:
The [tex]\( y \)[/tex]-intercept of the second function is [tex]\(-2\)[/tex].
Thus, the [tex]\( y \)[/tex]-intercept of the function with the greater rate of change is [tex]\( -2 \)[/tex].
