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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

1 answer

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