Question

For the function g(x)=−∣x−2∣+1, which of the following statements is true about its graph?

Answer 1

The graph of g(x) = -|x-2|+1 is a horizontal line at some negative value.

Step-by-step explanation:

The graph of the function g(x) = -|x-2|+1 is a horizontal line at some negative value.

The absolute value function |x-2| takes the distance from x to 2 and ignores the sign.

When x is less than 2, the expression inside the absolute value becomes negative, resulting in -1 times a negative number, which is positive.

When x is greater than or equal to 2, the expression inside the absolute value becomes non-negative, resulting in -1 times a non-negative number, which is negative.

Therefore, the graph of g(x) starts at 1 and decreases horizontally as x increases from 2.

Answer 2

The graph of the function g(x)=-|x-2|+1 is a V-shaped graph with a vertex at the point (2, 1), and it opens downwards. None of the given statements are true as it is not a straight or horizontal line.

Step-by-step explanation:

The function g(x)=−|x−2|+1 represents a V-shaped graph where the absolute value function creates two linear segments with opposite slopes. The negative sign outside the absolute value inverts the usual V-shape, making the graph open downward instead of upward. This function is not a straight line; instead, its graph is V-shaped, where the vertex (the point where the graph changes direction) is at (2, 1). Since the function inside the absolute value is x−2, the vertex is the point where the input to the absolute value is zero, which occurs at x=2. The plus 1 outside of the absolute value shifts the graph up one unit. Therefore, the correct statement about the graph of this function is:

a. It is a straight line with negative slope. Incorrect
b. It is a straight line with positive slope. Incorrect
c. It is a horizontal line at some negative value. Incorrect
d. It is a horizontal line at some positive value. Incorrect

None of the given options accurately describes the graph of the function g(x). Instead, its graph is a V-shaped with a vertex at the point (2, 1), opening downwards due to the negative sign in front of the absolute value.