Question

What is the function rule? Write the rule in function notation, using the slope-intercept form of a line.

Answer 1

Step 1: Write out the formula for the equation of a line between two points (x1,y1) and (x2,y2):

The equation of a line passing through points (x1,y1) and (x2,y2) is given by:

`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)_{}`

Step 2: Write out the coordinates of two points on the given line.

From the graph, we can see that two points on the line are given by:

`(0,3)\text{ and }(3,1)`

In this case,

`\begin{gathered} x_1=0 \\ y_1=3 \\ x_2=3 \\ y_2=1 \end{gathered}`

Step 3: Substitute the values into the formula to find the equation of the line:

`\begin{gathered} (y-3)/(1-3)=(x-0)/(3-0) \\ \text{Subtracting the numbers, we have:} \\ (y-3)/(-2)=(x)/(3) \\ \text{Cross-multiplying, we have:} \\ 3(y-3)=-2x \\ \text{Expanding the left side, we have:} \\ 3y-9=-2x \\ \text{Adding 9 to both sides, we have} \\ 3y=-2x+9 \end{gathered}`

Hence,

`y=(-2x+9)/(3)`

Therefore, the function rule is given by:

y = (-2x + 9)/3