Which is one of the transformations applied to the graph of \( f(x)=x^{2} \) to change it into the graph of \( g(x)=4 x^{2}+24 x+ \) The graph of \( f(x)=x^{2} \) is widened. The graph of \( f(x)=x^{2} \) is shifted left 3 units. The graph of \( f(x)=x^{2} \) is shifted up 30 units. The graph of \( f(x)=x^{2} \) is reflected over the \( x \)-axis.

To transform the graph of \( f(x) = x^{2} \) into \( g(x) = 4x^{2} + 24x + C \) (where \( C \) is any constant), we first need to complete the square or rewrite \( g(x) \) in vertex form to identify the transformations. The factor of \( 4 \) indicates that the graph of \( f(x) = x^2 \) is widened by a factor of \( 1/4 \). The term \( 24x \) suggests that the graph is shifted left (the zero of this quadratic can be calculated). Since this transformation involves expanding the graph horizontally but does not cause vertical shifts, the correct answer is that the graph is widened. Next, understanding how equations of this type affect the graph is key. Quadratics in standard form (\( ax^2 + bx + c \)) can be transformed into vertex form, greatly aiding in visualizing how the basic parabola shifts and stretches. Techniques like completing the square can make these transformations easier to manage.