Answer:
y = x + 7
Explanation:
To model the arch using the vertex form, we need to identify the vertex and the value of "a" in the quadratic equation.
First, let's visualize the arch. We have an arch with two ends, which are 7 feet apart. The vertex is the midpoint of the arch, which is the highest point. The vertex will have the coordinates (x, y).
Given that the ends of the arch are 7 feet apart, we can calculate the x-coordinate of the vertex by finding the midpoint of the arch. The midpoint is simply the average of the x-coordinates of the ends:
x = (0 + 7) / 2 = 3.5
To find the y-coordinate of the vertex, we need to know the height of the arch at the vertex. We know that one of the footholds is 2 feet off the ground and 2 feet from the end of the arch. This means that at a distance of 3.5 - 2 = 1.5 feet from the vertex, the arch is also 2 feet off the ground. Therefore, the y-coordinate of the vertex is 2.
Now that we have the vertex (3.5, 2), we can write the equation in vertex form:
y = a(x - h)^2 + k
where (h, k) represents the vertex. Plugging in the values, we have:
y = a(x - 3.5)^2 + 2
To find the value of "a", we need another point on the arch. Let's use one of the ends of the arch. The coordinates of the end points are (0, 0) and (7, 0). We can substitute these coordinates into the equation to solve for "a".
When x = 0, y = 0:
0 = a(0 - 3.5)^2 + 2
0 = a(3.5)^2 + 2
0 = a(12.25) + 2
0 = 12.25a + 2
-2 = 12.25a
a = -2 / 12.25
Simplifying the value of a, we have:
a ≈ -0.163
Now we can rewrite the equation with the value of a:
y = -0.163(x - 3.5)^2 + 2