Question

1. the line which is the graph of $y = 2x - 4$ is transformed by the rule $(x, y) \to (-x, -y)$. what is the slope of the image? click submit after typing your answer and a sample response will show. you will not be able to edit your response after submitting.

Explanation:

Step1: Recall transformation rule

The transformation rule is \((x, y) \to (-x, -y)\), which is a rotation about the origin by \(180^\circ\) or a reflection over both axes. For a linear function \(y = mx + b\), after this transformation, we can find the new equation.

Step2: Apply transformation to the line

Take a general point \((x, y)\) on \(y = 2x - 4\). After transformation, the new point is \((-x, -y)\). So we substitute \(x\) with \(-x\) and \(y\) with \(-y\) in the original equation:
\(-y = 2(-x) - 4\)

Step3: Solve for the new \(y\)

Multiply both sides by \(-1\) to solve for \(y\):
\(y = -2(-x) + 4\)
Simplify the right - hand side: \(y = 2x+4\)

Step4: Identify the slope

The equation of a line in slope - intercept form \(y = mx + b\) has a slope of \(m\). For the line \(y = 2x + 4\), the slope \(m = 2\).

Answer:

2