Question

write the point - slope form of the equation of the line satisfying each of the following conditions. then use the point - slope form to write the slope - intercept form of the equation in function notation. slope = $-\frac{2}{3}$, passing through $(9, - 6)$ the point - slope form of the equation is \\(\square\\). (simplify your answer. type an equation. use integers or fractions for any numbers in the equation.) let f be the name of the function. the slope - intercept form of the equation written in function notation is \\(\square\\). (simplify your answer. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Recall point - slope formula

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line. We are given that \(m =-\frac{2}{3}\) and the point \((x_1,y_1)=(9,-6)\).
Substitute these values into the point - slope formula:
\(y-(-6)=-\frac{2}{3}(x - 9)\)
Simplify the left - hand side: \(y + 6=-\frac{2}{3}(x - 9)\)

Step2: Convert point - slope to slope - intercept form

First, expand the right - hand side of the point - slope equation.
Using the distributive property \(a(b - c)=ab - ac\), where \(a =-\frac{2}{3}\), \(b=x\) and \(c = 9\), we get:
\(y+6=-\frac{2}{3}x+(-\frac{2}{3})\times(-9)\)
Calculate \((-\frac{2}{3})\times(-9)\): \((-\frac{2}{3})\times(-9)=6\)
So the equation becomes \(y + 6=-\frac{2}{3}x+6\)
Subtract 6 from both sides to solve for \(y\):
\(y=-\frac{2}{3}x+6 - 6\)
Simplify the right - hand side: \(y=-\frac{2}{3}x\)

Step3: Write in function notation

If \(f\) is the name of the function, then \(y = f(x)\). So the slope - intercept form in function notation is \(f(x)=-\frac{2}{3}x\)

Answer:

The point - slope form of the equation is \(y + 6=-\frac{2}{3}(x - 9)\)
The slope - intercept form of the equation written in function notation is \(f(x)=-\frac{2}{3}x\)