Question

mathematics ii a thayer fall 25 analyzing functional relationships the computer rendering of a mural in a towns square uses the function represented in the table to define the outline of a mountain in the towns logo, where x is the distance in feet from the edge of the mural and f(x) is the distance from the ground in feet. x: 4, 8, 10, 12, 20; f(x): -16, 0, 8, 16, -16. how can the point (12, 16) be explained? options: 1. the highest point of the mountain defined by the function is 12 feet. 2. the width of the base of the mountain defined by the function is 12 feet. 3. the width of the base of the mountain defined by the function is 16 feet. 4. the highest point of the mountain defined by the function is 16 feet.

Explanation:

To explain the point \((12, 16)\) in the context of the function \(f(x)\) (where \(x\) is the distance from the edge of the mural in feet and \(f(x)\) is the distance from the ground in feet), we analyze the coordinates:

• The \(x\)-value is \(12\), which represents the horizontal distance from the edge of the mural.
• The \(y\)-value (or \(f(x)\)-value) is \(16\), which represents the vertical distance from the ground.

Looking at the options, we need to match what \((12, 16)\) means. The option "The highest point of the mountain defined by the function is 16 feet" is incorrect because \(x = 12\) is a horizontal distance, not related to the height being "12 feet" in that way. The option "The width of the base of the mountain defined by the function is 16 feet" is incorrect as width would relate to a horizontal span, not the \(y\)-value. The option "The width of the base of the mountain defined by the function is 12 feet" is incorrect for similar reasons (width is horizontal, not \(x = 12\) defining width).

The correct interpretation is that at \(x = 12\) (horizontal distance from the edge), the height (distance from the ground) \(f(x)=16\) represents the highest point? Wait, no—wait, let's re - evaluate. Wait, the point \((12, 16)\) has \(x = 12\) (distance from edge) and \(f(x)=16\) (distance from ground). So the option "The highest point of the mountain defined by the function is 16 feet" is saying that the maximum height (the \(f(x)\)-value) of the mountain (the function) is 16 feet, which is consistent with \(f(12)=16\) being a peak (since other points like \(x = 4\) have \(f(4)=- 16\), \(x = 8\) has \(f(8)=0\), \(x = 10\) has \(f(10)=8\), \(x = 12\) has \(f(12)=16\), \(x = 20\) has \(f(20)=-16\) – so the function seems to have a maximum at \(x = 12\) with \(f(x)=16\), meaning the highest point of the mountain (the function's graph representing the mountain) is 16 feet above the ground. Wait, no, the option says "The highest point of the mountain defined by the function is 16 feet" – actually, the correct option is "The highest point of the mountain defined by the function is 16 feet" (the third option? Wait, let's list the options properly:

Option 1: "The highest point of the mountain defined by the function is 12 feet." (Incorrect, since \(f(12)=16\), not 12)

Option 2: "The width of the base of the mountain defined by the function is 12 feet." (Incorrect, width is horizontal, not related to \(x = 12\) as width)

Option 3: "The highest point of the mountain defined by the function is 16 feet." (Correct, because \(f(12)=16\) is the maximum value of \(f(x)\) among the given points, so it's the highest point of the mountain (the function's graph) with height 16 feet)

Option 4: "The width of the base of the mountain defined by the function is 16 feet." (Incorrect, width is horizontal, not 16)

Wait, maybe I misread the options. Let's re - check the image description:

The options are:

1. "The highest point of the mountain defined by the function is 12 feet."

1. "The width of the base of the mountain defined by the function is 12 feet."

1. "The highest point of the mountain defined by the function is 16 feet."

1. "The width of the base of the mountain defined by the function is 16 feet."

Since \(f(12) = 16\), and \(x = 12\) is the horizontal distance, the \(y\)-value (16) is the height. So the highest point of the mountain (the function's graph, which represents the mountain) has a height of 16 feet. So the correct option is "The highest point of the mountain defined by the function is 16 feet."

Answer:

The highest point of the mountain defined by the function is 16 feet.