Question

the graph of \\(y=f(x)\\) is shown below.
draw the graph of \\(y=-f(x)\\).

Explanation:

Identify key points of the original graph

The original graph of \(y = f(x)\) is a piecewise linear function. We can identify its key vertices and endpoints from the grid:

• Left endpoint: \((-7, -3)\)
• First turning point: \((-2, 2)\)
• Right endpoint: \((0, 0)\)

Apply the transformation rule

To graph \(y = -f(x)\), we reflect the graph of \(y = f(x)\) vertically across the \(x\)-axis. This transformation negates the \(y\)-coordinate of every point on the graph while keeping the \(x\)-coordinate the same:

$$(x, y) ightarrow (x, -y)$$

Calculate the transformed key points

Applying the reflection rule \((x, -y)\) to our identified key points:

• \((-7, -3)

ightarrow (-7, 3)\)

• \((-2, 2)

ightarrow (-2, -2)\)

• \((0, 0)

ightarrow (0, 0)\)

Connect the transformed points

Draw straight line segments connecting the new key points in order:

• A line segment from \((-7, 3)\) to \((-2, -2)\).
• A line segment from \((-2, -2)\) to \((0, 0)\).

Answer:

To draw the graph of \(y = -f(x)\), reflect the original graph vertically across the \(x\)-axis by negating the \(y\)-coordinates of its key points:

• The point \((-7, -3)\) becomes \((-7, 3)\).
• The point \((-2, 2)\) becomes \((-2, -2)\).
• The point \((0, 0)\) remains \((0, 0)\).

Plot these points and connect them with straight line segments: from \((-7, 3)\) to \((-2, -2)\), and from \((-2, -2)\) to \((0, 0)\).