The main cable of a suspension bridge forms a parabola, described by the equation [tex]y=a(x-h)^2+k[/tex].
| x | 0 | 52.5 | 105 | 157.6 | 210 |
| --- | -- | ---- | --- | ----- | --- |
| y | 27 | 12 | 7 | 12 | 27 |
[tex]y=[/tex] height in feet of the cable above the roadway
[tex]x=[/tex] horizontal distance in feet from the left bridge support
[tex]a=[/tex] a constant
([tex]h, k[/tex]) = vertex of the parbola
What is the vertex of the parabola?
Answer (2)
Recognize the symmetry of the parabola from the given data.
Calculate the x-coordinate of the vertex h by averaging symmetric x values: h = 2 0 + 210 = 105 .
Identify the y-coordinate of the vertex k from the table: when x = 105 , y = 7 .
State the vertex of the parabola: ( 105 , 7 ) .
Explanation
Understanding the Problem We are given a parabola in the form y = a ( x − h ) 2 + k , where ( h , k ) is the vertex. We have a table of x and y values. Our goal is to find the vertex ( h , k ) .
Finding the Axis of Symmetry Notice that the y values are symmetric around some x value. Specifically, when x = 0 , y = 27 , and when x = 210 , y = 27 . Also, when x = 52.5 , y = 12 , and when x = 157.6 , y = 12 . This suggests that the vertex is located at the midpoint of the x values for which the y values are equal.
Calculating the x-coordinate of the Vertex To find the x -coordinate of the vertex ( h ), we can take the average of the x values that give the same y value. For example, using the points where y = 27 , we have h = 2 0 + 210 = 105 . Using the points where y = 12 , we have h = 2 52.5 + 157.6 = 2 210.1 = 105.05 . Since the value 157.6 is approximate, we will assume that the vertex is at h = 105 .
Finding the y-coordinate of the Vertex Now that we have the x -coordinate of the vertex, we can find the y -coordinate ( k ) by looking at the table. When x = 105 , y = 7 . Therefore, the vertex is ( 105 , 7 ) .
Final Answer Thus, the vertex of the parabola is ( h , k ) = ( 105 , 7 ) .
Examples
Understanding parabolas can help design suspension bridges, where the main cable follows a parabolic path. Knowing the vertex (lowest point) is crucial for determining cable length and bridge stability. Similarly, satellite dishes and telescope mirrors use parabolic shapes to focus signals or light, with the vertex being the optimal point for the receiver or sensor. In sports, a ball's trajectory (like a basketball or soccer ball) often approximates a parabola, and understanding the vertex helps athletes optimize their throws or kicks for maximum distance or height. These applications demonstrate how understanding parabolas and their vertices is essential in engineering, science, and sports.
The vertex of the parabola, which represents the main cable of the suspension bridge, is located at (105, 7). This point signifies the lowest height of the cable above the roadway. Understanding the vertex location is crucial for analyzing the cable's shape and structural integrity.
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