Question

Use the image to answer the question.

An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 2 comma 4 right parenthesis, left parenthesis 4 comma 8 right parenthesis, and left parenthesis 5 comma 10 right parenthesis.

The proportional relationship is shown on the graph. Derive the equation of the line y=mx through the origin.

(1 point)
Responses

y=4x
y equals 4 x

y=2x
y equals 2 x

y=12x
y equals Start Fraction 1 over 2 End Fraction x

y=8x

Answer 1

To derive the equation of the line in the form \( y = mx \) that passes through the origin, we need to determine the slope \( m \) of the line.

To find the slope, we can use any two points on the line. For example, using the points \( (2, 4) \) and \( (4, 8) \):

1. Calculate the change in \( y \) (rise):
\[
\Delta y = y_2 - y_1 = 8 - 4 = 4
\]

2. Calculate the change in \( x \) (run):
\[
\Delta x = x_2 - x_1 = 4 - 2 = 2
\]

3. Calculate the slope \( m \):
\[
m = \frac{\Delta y}{\Delta x} = \frac{4}{2} = 2
\]

Now, since the line is proportional and passes through the origin, the equation of the line is:
\[
y = 2x
\]

So the correct response is:
**y = 2x**