Question

Find the rational function which represents the graph. If you could help with both sections that would be amazing

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given points on the graph

`\begin{gathered} \text{ Points are given in the form of }(x,y) \\ \therefore\text{ points are:} \\ (x_1,y_1)=(2,0) \\ (x_2,y_2)=(0,-2) \end{gathered}`

STEP 2: Write the slope-intercept form of the equation of a line

`\begin{gathered} y=mx+b \\ \text{where m is the slope and b is the y-intercept} \end{gathered}`

STEP 3: Write the formula to get the equation of a line using two given points

`(y-y_1)=m(x-x_1)`

STEP 4: We use the given points to get the slope

`m=(y_2-y_1)/(x_2-x_1)`

STEP 5: Substitute the given points into the formula in step 4 to get the slope

`m=(-2-0)/(0-2)=(-2)/(-2)=1`

STEP 6: Since we have a slope and two points, we use the formula in step 3 to get the function that represents the line

`\begin{gathered} (y-y_1)=m(x-x_1) \\ U\sin g\text{ point }(2,0) \\ (y-0)=1(x-2) \\ y=1(x-2) \\ y=x-2 \end{gathered}`

STEP 7: We get the rational function of the line using the given hole

Hence, the rational function representing the line with the given hole is:

`(y)/(y-3)=(x-2)/(x-5)`