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Math  /  Algebra

QuestionThe given figure shows a transformation of the graph of f(x)=∣x∣f(x)=|x|. Write the equation for the transformed graph.
The equation is y=y= □\square

Studdy Solution

STEP 1

1. The original function is f(x)=∣x∣ f(x) = |x| .
2. The transformed graph has its vertex at (0, 6).
3. The transformed graph passes through the points (3, 0) and (-3, 0).
4. The transformation involves a vertical stretch and a vertical shift.

STEP 2

1. Determine the vertical shift of the graph.
2. Determine the vertical stretch factor.
3. Write the equation of the transformed graph.

STEP 3

The vertex of the original graph f(x)=∣x∣ f(x) = |x| is at (0, 0). The vertex of the transformed graph is at (0, 6). This indicates a vertical shift of 6 units upwards.

STEP 4

The original graph passes through (1, 1) and (-1, 1). The transformed graph passes through (3, 0) and (-3, 0).
To find the vertical stretch factor, observe that the distance from the vertex to the x-intercepts is 3 units in the transformed graph. In the original graph, this distance is 1 unit. Therefore, the vertical stretch factor is 3.

STEP 5

The transformation includes a vertical stretch by a factor of 3 and a vertical shift up by 6 units. The equation of the transformed graph is:
y=3∣x∣+6 y = 3|x| + 6
The equation for the transformed graph is:
y=3∣x∣+6 y = \boxed{3|x| + 6}

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