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Interpret the key features of the graph what type of transformation occurred from the parent function f of x equals 4x squared there's a point at negative 2, 16 2, 16 -1, 4 1, 4 and 0, 0

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To interpret the key features of the graph and describe the transformation from the parent function \( f(x) = 4x^2 \), we can break down the information provided, specifically looking at the points mentioned: (-2, 16), (2, 16), (-1, 4), (1, 4), and (0, 0).

  1. Identifying the Parent Function: The parent function is \( f(x) = 4x^2 \). This is a parabola that opens upwards with a vertex at the origin (0,0).

  2. Analyzing Given Points:

    • The point (0, 0) confirms that the vertex remains at the origin.
    • The points (1, 4) and (-1, 4) indicate that when \( x = 1 \) and \( x = -1 \), the output \( f(x) \) is 4. This corresponds to \( f(1) = 4(1^2) = 4 \) and similarly for -1. It confirms that the shape of the parabola is maintained.
    • The points (2, 16) and (-2, 16) suggest that at \( x = 2 \) and \( x = -2 \), \( f(x) = 16\). This corresponds to \( f(2) = 4(2^2) = 16 \) and similarly for -2.

  3. Comparing with the Parent Function: The provided points reflect the parabolic shape of \( f(x) = 4x^2 \), but we need to determine if any transformations have occurred:

    • The vertex at (0, 0) indicates there is no vertical shift.
    • Points at \( (1, 4) \) and \( (-1, 4) \) suggest the graph rises rapidly due to the coefficient of \( 4 \) in front of \( x^2 \).

  4. Understanding Transformations: The presence of the points indicates that the graph maintains the same shape and orientation as the parent function \( f(x) = 4x^2 \) but may have undergone a horizontal stretch or compression.

    • Vertical Stretch: The graph of \( f(x) = 4x^2 \) is vertically stretched compared to \( g(x) = x^2 \). The coefficient of 4 means that for any given \( x \), the \( y \)-value of the transformed graph is four times that of the basic \( x^2 \) graph.

  5. Summary: The key features of the observed graph suggest that:

    • The graph is a vertically stretched version of the parent function \( f(x) = x^2 \) due to the factor of 4.
    • No shifts or reflections have occurred, meaning the vertex remains at the origin.
    • The parabolic shape is preserved as indicated by the symmetry of points around the y-axis.

In conclusion, the function appears to be \( f(x) = 4x^2 \) and has undergone a vertical stretch without any additional transformations.