the graph of a linear function passes through the points (4,2) and (12,0). write an equation for the function.

Explanation:

Step1: Calculate the slope

The slope ( m ) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by ( m=\frac{y_2 - y_1}{x_2 - x_1} ). Here, ((x_1, y_1)=(4, 2)) and ((x_2, y_2)=(12, 0)). So, ( m=\frac{0 - 2}{12 - 4}=\frac{-2}{8}=-\frac{1}{4} ).

Step2: Use point - slope form

The point - slope form of a linear equation is ( y - y_1=m(x - x_1) ). We can use the point ((4, 2)) and ( m =-\frac{1}{4} ). Substituting these values in, we get ( y - 2=-\frac{1}{4}(x - 4) ).

Step3: Simplify the equation

Expand the right - hand side: ( y - 2=-\frac{1}{4}x + 1 ). Then, add 2 to both sides: ( y=-\frac{1}{4}x+1 + 2 ), so ( y =-\frac{1}{4}x+3 ). We can also write it in standard form or other forms, but the slope - intercept form is ( y=-\frac{1}{4}x + 3 ) or in the form ( y-\ 2=-\frac{1}{4}(x - 4) ) or ( y=-\frac{1}{4}x+3 ). If we want to write it as ( y=-\frac{1}{4}x + 3 ), we can also rearrange it as ( y+\frac{1}{4}x=3 ), but the most common slope - intercept form is ( y =-\frac{1}{4}x+3 ).

Answer:

( y=-\frac{1}{4}x + 3 ) (or equivalent forms like ( y - 2=-\frac{1}{4}(x - 4) ))