Questions: Use transformations to graph the given function. g(x) = 1/2 x^2

Transcript text: Use transformations to graph the given function. \[ g(x)=\frac{1}{2} x^{2} \]

Solution

Solution Steps

Step 1: Identify the parent function

The parent function is f(x) = x². This is a standard parabola with its vertex at the origin (0,0).

Step 2: Determine the transformation

The given function is g(x) = (1/2)x². The coefficient 1/2 in front of the x² term indicates a vertical compression or shrinking. This means the parabola will be wider than the parent function.

Step 3: Graph the transformed function

Start with the parent function f(x) = x². Key points are (0,0), (1,1), (-1,1), (2,4), (-2,4). For the transformed function g(x) = (1/2)x², multiply the y-coordinates of the parent function's key points by 1/2. This gives us the points (0,0), (1,1/2), (-1,1/2), (2,2), (-2,2). Plotting these points and drawing a smooth curve through them gives the graph of g(x).

Final Answer:

The graph of g(x) is a parabola vertically compressed by a factor of 1/2 compared to the parent function f(x) = x². The graph should open upwards with the vertex at (0,0) and pass through points like (1, 0.5), (-1, 0.5), (2, 2), and (-2, 2). It is wider than the graph of x².

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