Question

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Changing Slope and y-intercept Quick Check
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Use the image to answer the question.

A solid line is plotted in quadrants 3 and 4 of a coordinate plane. The values on the x-axis range from negative 2 to 2 in increments of 1. The values on the y-axis range from negative 3 to 0 in increments of 1. A solid line begins in quadrant 3 sloping downward from left to right and descends through quadrant 4. The line passes through left parenthesis negative 2 comma negative 1 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma negative 3 right parenthesis.

Predict the equation of the line shown if it were shifted up 8 units.

(1 point)
Responses

y=−0.5x+6
y equals negative 0.5 x plus 6

y=−0.5x−8
y equals negative 0.5 x minus 8

y=−0.5x−2
y equals negative 0.5 x minus 2

y=−0.5x+8
y equals negative 0.5 x plus 8
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Answer 1

To predict the equation of the line when it is shifted up 8 units, you first need to determine the original equation of the line.

From the points given:

1. \((-2, -1)\)
2. \((0, -2)\)
3. \((2, -3)\)

You can calculate the slope (\(m\)) of the line using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Taking the points \((0, -2)\) and \((-2, -1)\):

\[
m = \frac{-1 - (-2)}{-2 - 0} = \frac{1}{-2} = -0.5
\]

Now, you can use the point-slope form of the line equation, \(y - y_1 = m(x - x_1)\). Using the point \((0, -2)\):

\[
y - (-2) = -0.5(x - 0) \\
y + 2 = -0.5x \\
y = -0.5x - 2
\]

This is the original equation of the line: \(y = -0.5x - 2\).

Now, when this line is shifted up 8 units, you add 8 to the entire equation:

\[
y = -0.5x - 2 + 8 \\
y = -0.5x + 6
\]

So, the equation of the line after being shifted up 8 units is:

\[
\boxed{y = -0.5x + 6}
\]