Math / Geometry

Question(a) The graph of $y=f(x)$ is shown. Draw the graph of $y=f\left(\frac{1}{2} x\right)$ .

Studdy Solution

STEP 1

1. The function $y = f(x)$ is continuous and defined for the given range.
2. The transformation involves a horizontal scaling of the graph.

STEP 2

1. Understand the effect of the transformation $y = f\left(\frac{1}{2} x\right)$ .
2. Identify key points on the original graph.
3. Apply the transformation to these key points.
4. Sketch the transformed graph.

STEP 3

Understand the effect of the transformation $y = f\left(\frac{1}{2} x\right)$ :
The transformation $y = f\left(\frac{1}{2} x\right)$ represents a horizontal stretch of the graph by a factor of 2. This means that each x-coordinate on the original graph will be multiplied by 2.

STEP 4

Identify key points on the original graph:
For the given V-shaped graph, the vertex is at $(-2, -2)$ . Other key points can be identified on the arms of the V, such as $(-4, 0)$ and $(0, 0)$ .

STEP 5

Apply the transformation to these key points:
- For the vertex $(-2, -2)$ , the transformed point is $(-2 \times 2, -2) = (-4, -2)$ . - For the point $(-4, 0)$ , the transformed point is $(-4 \times 2, 0) = (-8, 0)$ . - For the point $(0, 0)$ , the transformed point is $(0 \times 2, 0) = (0, 0)$ .

STEP 6

Sketch the transformed graph:
- Plot the transformed points: $(-8, 0)$ , $(-4, -2)$ , and $(0, 0)$ . - Connect these points to form the V-shape, ensuring the arms extend symmetrically from the vertex $(-4, -2)$ .
The graph of $y = f\left(\frac{1}{2} x\right)$ is a horizontally stretched version of the original graph, with the vertex now at $(-4, -2)$ .

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