Question

Compare the graph of g(x) = -3x2 - 5 with the graph of f(x) = x2

Answer 1

1. The graph of
`\( g(x) = -3x^2 - 5 \)` is a downward-facing parabola, reflecting a vertical stretch by a factor of 3 and a vertical translation downward by 5 units compared to the graph of
`\( f(x) = x^2 \).`

Step-by-step explanation:

The function
`\( f(x) = x^2 \)` represents a standard upward-facing parabola with its vertex at the origin. The given function
`\( g(x) = -3x^2 - 5 \)` is a transformation of
`\( f(x) \)`. The coefficient -3 indicates a vertical stretch by a factor of 3, causing the parabola to open downward. The constant term -5 represents a vertical translation downward by 5 units.

To understand the transformation, consider specific points. For instance, when
`\( x = 1 \), in \( f(x) \), \( f(1) = 1^2 = 1 \),` and in
`\( g(x) \), \( g(1) = -3(1)^2 - 5 = -8 \)`. This demonstrates the vertical stretch and translation in action.

The negative coefficient in
`\( g(x) \)` reflects the parabola's orientation, making it open downward. The vertical stretch by a factor of 3 amplifies the steepness of the graph, and the downward translation shifts the entire graph lower. Therefore, the comparison between the graphs of
`\( g(x) \)` and
`\( f(x) \)` indicates a transformed parabola that is wider, steeper, and shifted downward.

Answer 2

`g(x)=-3x^2-5`
`f(x)=x^2`

Let us put g(x) in the vertex form at first

`g(x)=-3(x-0)^2-5`

-3 means the graph of f(x) stretched vertically by a scale factor 3 and also reflected across the x-axis

-5 means the graph of f(x) is translated down by 5 units

Then f(x) is stretched vertically by a scale factor 3, reflected across the x-axis and translated 5 units down