Question

if f(x) is an even function and (6, 8) is one the points on the graph of f(x), which reason explains why (-6, 8) must also be a point on the graph?

since the function is even, the outputs of a negative x - value and a positive x - value are the same.

since the function is even, the outputs of a negative y - value and a positive y - value are the same.

the graph has rotational symmetry, so the point will be reflected across the y - axis.

the graph has rotational symmetry, so the point will be rotated 90 degrees about the origin.

Explanation:

1. Recall the definition of an even function: For an even function \( f(x) \), \( f(-x)=f(x) \) for all \( x \) in the domain. This means that if \( (a, b) \) is on the graph (so \( f(a) = b \)), then \( f(-a)=b \), so \( (-a, b) \) is also on the graph. In other words, the output (y - value) for \( -x \) is the same as the output for \( x \).
2. Analyze the options:

• Option 1: "Since the function is even, the outputs of a negative x - value and a positive x - value are the same." This matches the definition of an even function. If \( x = 6 \) gives \( f(6)=8 \), then \( x=- 6 \) gives \( f(-6)=f(6) = 8 \), so \( (-6,8) \) is on the graph.
• Option 2: Even functions are about the relationship between \( f(x) \) and \( f(-x) \), not about negative and positive y - values. So this option is incorrect.
• Option 3: Even functions have reflection symmetry about the y - axis, not rotational symmetry. Rotational symmetry is a property of odd functions (rotation by 180 degrees about the origin) or other types of functions, not even functions. So this option is incorrect.
• Option 4: Even functions do not have 90 - degree rotational symmetry about the origin. This is not a property of even functions. So this option is incorrect.

Answer:

A. Since the function is even, the outputs of a negative x - value and a positive x - value are the same.