Analyze the graph of f(x)

Question

. Which of the following is the range of the function f(x)=−|x−3|+78
?(1 point)
Responses

[−78,∞)
left bracket negative 7 eighths comma infinity right parenthesis

[78,∞)
left bracket 7 eighths comma infinity right parenthesis

(−∞,78]
left parenthesis negative infinity comma 7 eighths right bracket

(−∞,−78]

Answer 1

To analyze the function \( f(x) = -|x - 3| + 78 \), we first need to understand the behavior of the absolute value function, as well as the transformations applied to it.

1. Basic Structure:

   • The function \( g(x) = |x - 3| \) is a V-shaped graph with its vertex at (3, 0). It opens upwards.

2. Transformation:

   • The negative sign in front of the absolute value, \( -|x - 3| \), reflects the graph of \( g(x) \) over the x-axis, making it open downwards.
   • The \( + 78 \) moves the entire graph up by 78 units.

3. Vertex:

   • The vertex of the original function \( g(x) \) is at \( (3, 0) \). After the transformations, the vertex of \( f(x) \) will be at \( (3, 78) \).

4. Behavior:

   • Since the function opens downward (due to the negative sign), the maximum value of \( f(x) \) occurs at \( x = 3 \), yielding \( f(3) = 78 \).
   • As \( x \) moves away from 3 in either direction (left or right), the value of \( f(x) \) decreases indefinitely due to the decreasing behavior of the function.

5. Range:

   • The maximum value of \( f(x) \) is 78 (at the vertex), and as we move away from this point, the function values decrease, approaching negative infinity. Hence, the range starts from negative infinity and goes up to 78.

Thus, the range of the function \( f(x) = -|x - 3| + 78 \) can be expressed as:

\[ (-\infty, 78] \]

The correct response is: \( (-\infty, 78] \).