Question

Please help me with this graph.

Answer 1

Answer:

$y=\frac{1}{2} x-1$

Step-by-step explanation:

Please refer to the attached image.

Recall that in order to find the equation of a line parallel to another one, one needs to have the actual slope of the original line. This is because one will need to give the same slope in the equation for the new line.

Let's now try to extract the slope of the first line by looking at its graph. Recall as well that the formula for the slope uses information from two points on the plane $(x_1,y_1) ; (x_2,y_2)$ through which the line passes. We then try to find two points with easily identifiable coordinates. Please find two such points marked with red dots in the attached image. Their coordinates are also shown: (0,4) and (10,9).

Now let's use the formula for the slope (m) given two points  $(x_1,y_1) ; (x_2,y_2)$

$m=\frac{y_2-y_1}{x_2-x_1}$

If we assign $(x_1,y_1)=(0,4)$, and $(x_2,y_2)=(10,9)$ the slope becomes:

$m=\frac{y_2-y_1}{x_2-x_1}\\m=\frac{9-4}{10-0}\\m=\frac{5}{10}\\m=\frac{1}{2}$

Then the parallel line we are going to build should also have slope "1/2"

Now, the next step is to create the equation of the new line using the slope m = 1/2 and also using the point (4,1) where it should be passing. So we use the "point-slope" formula to create the line:

$y-y_0=m\,(x-x_0)$

where $(x_0,y_0)$ is the coordinate pair of thepont we want the line to go through (in our case the point (4,1)). Then:

$y-y_0=m\,(x-x_0)\\\\y-1=\frac{1}{2} (x-4)\\y-1=\frac{1}{2} x-2\\y=\frac{1}{2} x-2+1\\y=\frac{1}{2} x-1$ which is the third option among your list of answers.