Question

question 8 of 10
f(x) = x². what is g(x)?
graph of f(x) = x² (blue parabola opening up) and g(x) (red parabola opening down) on a coordinate plane
text description for graph
a. g(x) = x² - 2
b. g(x) = -x² - 2
c. g(x) = -2x²
d. g(x) = -x²

Explanation:

Step1: Analyze reflection

The graph of \( f(x) = x^2 \) is a parabola opening upwards. The graph of \( g(x) \) opens downwards, so there is a reflection over the x - axis. This means the coefficient of \( x^2 \) in \( g(x) \) should be negative. So we can eliminate option A (\( g(x)=x^2 - 2 \)) which has a positive coefficient for \( x^2 \).

Step2: Analyze vertical shift

The vertex of \( f(x)=x^2 \) is at \( (0,0) \). The vertex of \( g(x) \) is at \( (0, - 2) \) (since it is shifted down by 2 units from the origin). The general form of a parabola is \( y = a(x - h)^2+k \), where \( (h,k) \) is the vertex. For \( g(x) \), \( h = 0 \), \( k=-2 \) and \( a=-1 \) (because of reflection). So the equation should be \( g(x)=-x^2-2 \). Let's check the other options: Option C (\( g(x) = - 2x^2 \)) has a vertex at \( (0,0) \) (since \( k = 0 \)), which does not match the vertex of \( g(x) \) in the graph. Option D (\( g(x)=-x^2 \)) has a vertex at \( (0,0) \), which also does not match the vertex of \( g(x) \) (which is at \( (0,-2) \)).

Answer:

B. \( g(x)=-x^2 - 2 \)