Question

How do you write a function rule for $ x = 2,4,6 $ and $ y = 1,0, - 1 $ ?

Answer 1

Hint: Here in this question, we have to find the function where the points are given. In the question we have “write function rule” means the relation of a function where it contains both the function x and y. So, by considering points and by the equation of line we calculate the function

Complete step-by-step answer:
By the equation we can determine the points that are coordinate points. Likewise, by the points we can find or calculate the function. Here in this question, we have the values of x and y. by using these points we have to determine the function.
Now consider the values of x i.e., $ x = 2,4,6 $ . The step length or step size for the value of x is 2. Therefore, the change in x is 2.
Now consider the values of y i.e., $ y = 1,0, - 1 $ . The step length or step size for the value of y is -1. Therefore, the change in y is -1.
Therefore the coordinate points are (2,1), (4,0) and (6,-1)
To determine the function rule, we consider the formula of the equation of line. The equation of line is given by $ (y - {y_0}) = m(x - {x_0}) $ . The m is the slope of the line. Where $ m = \dfrac{{dy}}{{dx}} $ , $ dy $ means change in y and $ dx $ means change in x. Therefore $ m = \dfrac{{dy}}{{dx}} = \dfrac{{ - 1}}{2} $ .
Now let we choose any one of the points as $ ({x_0},{y_0}) $ and substitute in the equation of a line. We consider (4,0) as $ ({x_0},{y_0}) $ . Substituting the values in the equation of line we have
 $ (y - {y_0}) = m(x - {x_0}) $
 $ \Rightarrow (y - 0) = \dfrac{{ - 1}}{2}(x - 4) $
On simplifying the above equation, we have
 $ \Rightarrow y = \dfrac{{ - 1}}{2}x + 2 $
Therefore, the function for the points $ x = 2,4,6 $ and $ y = 1,0, - 1 $ is $ y = \dfrac{{ - 1}}{2}x + 2 $ .
We can also verify the given equation by substituting the values. Let we take the value of as 2. Substituting the value of x in the equation we get
 $
  y = \dfrac{{ - 1}}{2}(2) + 2 \\
   \Rightarrow y = - 1 + 2 \\
   \Rightarrow y = 1 \;
  $
Hence verified.
So, the correct answer is “ $ y = \dfrac{{ - 1}}{2}(2) + 2 $ ”.

Note: A function rule describes how to convert an input (x) into an output (y) for a given function. The equation involves the parameter x and y. We can find the value of y by substituting the values of x. If we know the points that are coordinate points, we can find the equation by using the equation of line.