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The computer rendering of a mural in a town's square uses the function represented in the table to define the outline of a mountain in the town's 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.




$x$		$f(x)$

4		-16

8		0

10		8

12		16

20		-16






How can the point $(12,16)$ be explained?


AThe highest point of the mountain defined by the function is 12 feet.
BThe highest point of the mountain defined by the function is 16 feet.
CThe width of the base of the mountain defined by the function is 12 feet.
DThe width of the base of the mountain defined by the function is 16 feet.

Verified Answer :

  • The function \(f(x)\) represents the height of the mountain at distance \(x\) from the mural's edge.
  • The coordinate point \((12, 16)\) signifies that at a distance of \(12\) feet, the mountain reaches a height of \(16\) feet.
  • Comparing the values in the table, \(16\) is the highest value of \(f(x)\), identifying it as the peak height.
  • Thus, the point \((12, 16)\) shows that the highest point of the mountain is \(16\) feet.
  • Final answer: \(\boxed{\text{The highest point of the mountain defined by the function is 16 feet.}}\)

Explanation

  1. Analysis of the Problem and Data

The problem asks us to interpret the meaning of the coordinate point \((12, 16)\) in the context of a mountain's outline on a mural. We are given a function \(f(x)\) that maps the distance \(x\) from the edge of the mural to the height \(f(x)\) from the ground. The table provided lists several coordinate pairs \((x, f(x))\): \((4, -16)\), \((8, 0)\), \((10, 8)\), \((12, 16)\), and \((20, -16)\).

  1. Interpreting the Point \((12, 16)\)

The notation \((x, f(x)) = (12, 16)\) indicates that when the distance from the mural's edge is \(x = 12\) feet, the height of the mountain \(f(x)\) is \(16\) feet. By reviewing the heights listed in the table (\(-16, 0, 8, 16, -16\)), we observe that \(16\) feet is the largest vertical value recorded. Therefore, this point corresponds to the peak or the highest point of the mountain outline depicted by the function.

  1. Evaluating the Options and Conclusion

We evaluate the given options based on our interpretation. The coordinate \(x=12\) represents the location where the peak occurs, while \(f(x)=16\) represents the maximum height. Comparing this with the options provided, the statement "The highest point of the mountain defined by the function is \(16\) feet" accurately describes the meaning of the point \((12, 16)\) as the peak's maximum height.

Examples

In geography or civil engineering, mapping terrain requires plotting height data against distance. For instance, when designing a trail up a mountain, engineers identify key coordinates where the slope changes or reaches its summit. Understanding that a coordinate \((d, h)\) represents height \(h\) at a specific distance \(d\) allows planners to determine the mountain's peak, its steepness, and the best path for a safe and accessible trail.

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