Questions: Now find: Describe the transformations that produce g(x) from the parent graph f(x)=x^2 : Hint: This means to identify a, h, and k, and state what they do. g(x)=-2(x+3)^2-4

Solution Steps

To describe the transformations that produce \( g(x) \) from the parent graph \( f(x) = x^2 \), we need to identify the parameters \( a \), \( h \), and \( k \) in the vertex form of a quadratic function, which is \( g(x) = a(x-h)^2 + k \). The parameter \( a \) affects the vertical stretch or compression and reflection, \( h \) represents the horizontal shift, and \( k \) represents the vertical shift.

The function \( g(x) = -2(x+3)^2 - 4 \) can be compared to the vertex form of a quadratic function \( g(x) = a(x-h)^2 + k \). From this comparison, we identify the parameters:

• \( a = -2 \)
• \( h = -3 \)
• \( k = -4 \)

The identified parameters indicate the following transformations from the parent graph \( f(x) = x^2 \):

• The value \( a = -2 \) indicates a vertical stretch by a factor of \( 2 \) and a reflection over the x-axis.
• The value \( h = -3 \) indicates a horizontal shift \( 3 \) units to the left.
• The value \( k = -4 \) indicates a vertical shift \( 4 \) units down.

Final Answer