Given:
The graph of f passes through (-6,9).
It is perpendicular to the line that has an x-intercept of 8 and a y-intercept of -24.
To find:
The equation of the function f.
Solution:
The equation of line on which graph of f is perpendicular, is
[tex]\dfrac{x}{a}+\dfrac{y}{b}=1[/tex]
where, a and b are x and y intercepts respectively.
[tex]\dfrac{x}{8}+\dfrac{y}{-24}=1[/tex]
Multiply both sides by 24.
[tex]3x-y=24[/tex]
Slope intercept form is
[tex]y=3x-24[/tex]
Slope of this line is 3 and y-intercept is -24.
Product of slopes of two perpendicular lines is -1.
Let the slope of f is m. So,
[tex]m\times 3=-1[/tex]
[tex]m=-\dfrac{1}{3}[/tex]
Slope of m is -1/3 and it passes through (-6,9). So, the equation of function f is
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-9=-\dfrac{1}{3}(x-(-6))[/tex]
[tex]y-9=-\dfrac{1}{3}(x+6)[/tex]
[tex]y-9=-\dfrac{1}{3}x-2[/tex]
[tex]y=-\dfrac{1}{3}x-2+9[/tex]
[tex]y=-\dfrac{1}{3}x+7[/tex]
Put y=f(x).
[tex]f(x)=-\dfrac{1}{3}x+7[/tex]
Therefore, the required function is [tex]f(x)=-\dfrac{1}{3}x+7[/tex].
