Question

1. the equation $f(x) = (x - 1)^2 - 2$ represents $f(x)$ and the graph represents $h(x)$.

select whether each statement is true of false about the given functions.

	true	false
the minimum of $f(x)$ is less than the minimum of $h(x)$.	$\bigcirc$	$\bigcirc$

Explanation:

Step1: Analyze \( h(x) \)'s symmetry

For a parabola (quadratic function) in graph form, the line of symmetry is the vertical line through the vertex. The vertex of \( h(x) \) is at \( x = -3 \) (from the graph, the lowest point is at \( x = -3 \)). So the line of symmetry for \( h(x) \) is \( x = -3 \), so the first statement is True.

Step2: Find minima of \( f(x) \) and \( h(x) \)

• For \( f(x) = (x - 1)^2 - 2 \), it's a vertex - form quadratic \( y=a(x - h)^2 + k \), where vertex is \( (h,k)=(1, - 2) \). Since \( a = 1>0 \), the minimum value of \( f(x) \) is \( - 2 \) (at \( x = 1 \)).
• For \( h(x) \), from the graph, the vertex (minimum point) has a \( y \) - coordinate of \( - 1 \) (since it's 1 unit below the \( x \) - axis). So the minimum of \( f(x) \) is \( - 2 \), and the minimum of \( h(x) \) is \( - 1 \). Since \( - 2<-1 \), the minimum of \( f(x) \) is less than the minimum of \( h(x) \), so the second statement is True.

Answer:

1. "The line of symmetry of \( h(x) \) is \( x=-3 \)": True (select the circle in the "True" column for this row)
2. "The minimum of \( f(x) \) is less than the minimum of \( h(x) \)": True (select the circle in the "True" column for this row)