Question

graph the given functions f and g in the same rectangular coordinate system. describe how the graph of g is related to the graph of f.
$f(x) = -x^2$
$g(x) = -x^2 + 12$
use the graphing tool to graph the functions.
click to enlarge graph

Explanation:

Step1: Analyze the parent function

The function \( f(x) = -x^2 \) is a parabola opening downward with vertex at \((0, 0)\).

Step2: Analyze the transformed function

The function \( g(x) = -x^2 + 12 \) is also a parabola opening downward (since the coefficient of \( x^2 \) is -1, same as \( f(x) \)). To find the vertex of \( g(x) \), we note that for a quadratic function in the form \( y = ax^2 + bx + c \), the vertex is at \((-\frac{b}{2a}, c)\) when \( b = 0 \). Here, \( a=-1 \), \( b = 0 \), and \( c = 12 \), so the vertex of \( g(x) \) is at \((0, 12)\).

Step3: Describe the transformation

Comparing the vertex of \( f(x) \) (which is \((0, 0)\)) and the vertex of \( g(x) \) (which is \((0, 12)\)), we can see that the graph of \( g(x) \) is the graph of \( f(x) \) shifted vertically upward by 12 units. Since the coefficient of \( x^2 \) is the same for both functions (\(-1\)), there is no horizontal stretch, compression, or reflection other than the vertical shift.

Answer:

The graph of \( g(x)=-x^{2}+12 \) is the graph of \( f(x) = -x^{2} \) shifted vertically upward by 12 units. Both are downward - opening parabolas with the same shape (since the coefficient of \( x^{2} \) is the same), and the vertex of \( f(x) \) is \((0,0)\) while the vertex of \( g(x) \) is \((0,12)\).