x g x - 27 3 - 9 0 21 5 The table shows three values of x and their

Question

x g x - 27 3 - 9 0 21 5 The table shows three values of x and their corresponding values of g x , where g x = f x x + 3 and f is a linear function. What is the y -intercept of the graph of y = f x in the xy -plane?

Accepted Answer

Choice A is correct. It's given that the table shows values of x and their corresponding values of g x , where g x = f x x + 3 . It's also given that f is a linear function. It follows that an equation that defines f can be written in the form f x = m x + b , where m represents the slope and b represents the y -coordinate of the y -intercept 0 , b of the graph of y = f x in the xy -plane. The slope of the graph of y = f x can be found using two points, x 1 , y 1 and x 2 , y 2 , that are on the graph of y = f x , and the formula m = y 2 - y 1 x 2 - x 1 . Since the table shows values of x and their corresponding values of g x , substituting values of x and g x in the equation g x = f x x + 3 can be used to define function f . Using the first pair of values from the table, x = - 27 and g x = 3 , yields 3 = f - 27 - 27 + 3 , or 3 = f - 27 - 24 . Multiplying each side of this equation by - 24 yields - 72 = f - 27 , so the point - 27 , - 72 is on the graph of y = f x . Using the second pair of values from the table, x = - 9 and g x = 0 , yields 0 = f - 9 - 9 + 3 , or 0 = f - 9 - 6 . Multiplying each side of this equation by - 6 yields 0 = f - 9 , so the point - 9 , 0 is on the graph of y = f x . Substituting - 27 , - 72 and - 9 , 0 for x 1 , y 1 and x 2 , y 2 , respectively, in the formula m = y 2 - y 1 x 2 - x 1 yields m = 0 - - 72 - 9 - - 27 , or m = 4 . Substituting 4 for m in the equation f x = m x + b yields f x = 4 x + b . Since 0 = f - 9 , substituting - 9 for x and 0 for f x in the equation f x = 4 x + b yields 0 = 4 - 9 + b , or 0 = - 36 + b . Adding 36 to both sides of this equation yields 36 = b . It follows that 36 is the y -coordinate of the y -intercept 0 , b of the graph of y = f x . Therefore, the y -intercept of the graph of y = f x is 0 , 36 . Choice B is incorrect. 12 is the y -coordinate of the y -intercept of the graph of y = g x . Choice C is incorrect. 4 is the slope of the graph of y = f x . Choice D is incorrect. - 9 is the x -coordinate of the x -intercept of the graph of y = f x .

$x$	$g (x)$
$negative 27$	$3$
$negative 9$	$0$
$21$	$5$

Question x-g-x-