To solve this problem, we need to find the equation of a function [tex]\( f(x) = mx + b \)[/tex] where the graph of [tex]\( f \)[/tex] is perpendicular to the given line [tex]\( 4x - 9y - 27 = 0 \)[/tex] and has the same [tex]\( y \)[/tex]-intercept as this line.
### Step-by-Step Solution
1. Convert the given line equation to slope-intercept form:
The given line equation is [tex]\( 4x - 9y - 27 = 0 \)[/tex]. To convert this to the slope-intercept form [tex]\( y = mx + b \)[/tex], we need to solve for [tex]\( y \)[/tex]:
[tex]\[
4x - 9y - 27 = 0
\][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[
-9y = -4x + 27
\][/tex]
Divide by [tex]\(-9\)[/tex]:
[tex]\[
y = \frac{4}{9}x - 3
\][/tex]
Thus, the equation in slope-intercept form is [tex]\( y = \frac{4}{9}x - 3 \)[/tex].
2. Determine the slope and [tex]\( y \)[/tex]-intercept of the given line:
From the converted equation [tex]\( y = \frac{4}{9}x - 3 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\( \frac{4}{9} \)[/tex].
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( -3 \)[/tex].
3. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The original slope is [tex]\( \frac{4}{9} \)[/tex], so the slope of the perpendicular line is:
[tex]\[
- \frac{1}{\frac{4}{9}} = - \frac{9}{4}
\][/tex]
4. Find the equation of the function [tex]\( f(x) \)[/tex]:
The perpendicular line has the same [tex]\( y \)[/tex]-intercept as the given line, which is [tex]\( -3 \)[/tex].
Therefore, the equation of the function [tex]\( f(x) = mx + b \)[/tex] is:
[tex]\[
f(x) = - \frac{9}{4}x - 3
\][/tex]
Putting it into a simplified form:
[tex]\[
f(x) = -2.25x - 3
\][/tex]
Hence, the equation of the function whose graph is perpendicular to the given line and has the same [tex]\( y \)[/tex]-intercept is:
[tex]\[ f(x) = -2.25x + (-3) \][/tex]
### Final Answer
[tex]\[
f(x) = -2.25x - 3
\][/tex]
