Question

question 2 of 12 step 1 of 1 no time limit
graph the following function by considering how the function ( x ^ { 2 } ) has been shifted,
reflected, stretched or compressed.
g ( x ) = - 2 ( x - 2 ) ^ { 2 } - 3
answer 7 points
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points can be moved by dragging or using the arrow keys. points whose positions are
related to a moved point will be updated automatically. any lines or curves will be
drawn once all required points are plotted and will update whenever a point is moved.

Explanation:

Step1: Analyze the transformation of the function \(y = x^{2}\)

The general form of a quadratic function transformation is \(y=a(x - h)^{2}+k\). For the function \(g(x)=-2(x - 2)^{2}-3\), compared to \(y = x^{2}\):

• The factor \(a=-2\) indicates a vertical stretch by a factor of \(2\) and a reflection about the \(x\) - axis.
• The value \(h = 2\) indicates a horizontal shift of \(2\) units to the right.
• The value \(k=-3\) indicates a vertical shift of \(3\) units down.

Step2: Determine the vertex of the function \(g(x)\)

The vertex form of a parabola is \(y=a(x - h)^{2}+k\), and its vertex is \((h,k)\). For \(g(x)=-2(x - 2)^{2}-3\), the vertex is \((2,-3)\).

Step3: Find some additional points

Let's choose some \(x\) - values.
When \(x=1\):
\(g(1)=-2(1 - 2)^{2}-3=-2\times(-1)^{2}-3=-2 - 3=-5\)
When \(x=3\):
\(g(3)=-2(3 - 2)^{2}-3=-2\times1^{2}-3=-2 - 3=-5\)
When \(x=0\):
\(g(0)=-2(0 - 2)^{2}-3=-2\times4-3=-8 - 3=-11\)
When \(x=4\):
\(g(4)=-2(4 - 2)^{2}-3=-2\times4-3=-8 - 3=-11\)

Answer:

To graph \(g(x)=-2(x - 2)^{2}-3\):

• Plot the vertex at the point \((2,-3)\).
• Since the parabola is reflected about the \(x\) - axis (because \(a=-2<0\)) and vertically stretched by a factor of \(2\), and shifted \(2\) units right and \(3\) units down from \(y = x^{2}\).
• Use the additional points \((1,-5)\), \((3,-5)\), \((0,-11)\), \((4,-11)\) to sketch the parabola. The parabola opens downwards.