Question

How do you sketch the graph of `y=1/2x^2` and describe the transformation?

Answer 1

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Please read the explanation.

Explanation:

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We have the quadratic equation: `color(red)(y=f(x)=(1/2)*x^2`

Consider the Parent Function: `color(blue)(y=f(x)=x^2`

The General Form of a quadratic equation is:

`color(green)(y=f(x)=a*(x-h)^2+k`, where `color(red)((h,k)` is the Vertex

The graph of `color(blue)(y=f(x)=x^2`

pass through the origin `color(blue)((0,0)`

and the graphs uses both Quadrant-I and Quadrant-II.

Let us now look at the transformation of the graph.

Consider the format : `color(blue)(y=f(x)=a*x^2`

`color(red)([0 < | a | < 1]` results in a compression.

Since, in the problem,

`color(red)(y=f(x)=(1/2)*x^2`,

`color(green)(a=(1/2)`,

there will be a vertical compression toward the x-axis.

The graph is below:

Note that the graphs of

`y=f(x)=x^2` and

`y=f(x)=(1/2)*x^2`

are available together to examine the transformation.