Exercise 23 Page 185

Notice that the given two points, f( 2)= -2 and f( 1)= 1, are in function notation. To start, let's write these points as coordinate pairs. Remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f( x)&= y&⇔& ( x, y) f( -4)&= 2&⇔& ( -4, 2) f( 6)&= -3&⇔& ( 6, -3) Now, we are able to write an equation for function f in slope-intercept form. However, we cannot determine the y-intercept of the equation from the given points. Therefore, we will follow three steps.

1. We will first find the slope of the equation by using the Slope Formula.
2. Next, we will write the equation in point-slope form.
3. Finally, we will rearrange the equation to write it in slope-intercept form.

   Finding the Slope

   We know that the line of function f passes through the points ( -4, 2) and ( 6, -3). Let's substitute these points into the Slope Formula and find the slope.
   m = y_2-y_1/x_2-x_1

   SubstitutePoints

   Substitute ( -4,2) & ( 6,-3)

   m=-3- 2/6-( -4)

   ▼

   Simplify right-hand side

   SubNeg

   a-(- b)=a+b

   m=-3-2/6+4

   AddSubTerms

   Add and subtract terms

   m=-5/10

   ReduceFrac

   a/b=.a /5./.b /5.

   m=-1/2
   The slope of the line is - 12.

   Point-Slope Form

   We know the slope of the line and two points that are on the line. We can choose one of these points and write the equation of the line. Let's choose the point ( -4, 2). y- 2&= -1/2(x-( -4)) &⇓ y-2&=-1/2(x+4)

   Slope-Intercept Form

   Finally, we can write the equation of the line in slope-intercept form by isolating the y-variable in the equation.
   y-2=-1/2(x+4)

   Distr

   Distribute -1/2

   y-2=-1/2x-2

   AddEqn

   LHS+2=RHS+2

   y=-1/2x
   Therefore, the y-variable is isolated and we have our equation in slope-intercept form. f(x)=-1/2x