Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions.
x-intercepts −1 and 7
highest point has y-coordinate 4
vertical axis: y = a(x+1)(x-7)
if it has a highest point, it opens downward, so a < 0
The vertex is midway between the roots, at x=3
y(3) = a(4)(-4) = -16a = 4
So, find a and rearrange the equation to standard form
To find the standard equation of a parabola with a vertical axis, we need to determine its vertex form. The vertex form of a parabola equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Given the information that the x-intercepts are -1 and 7, we can conclude that the axis of symmetry is the line passing through the midpoint of the x-intercepts.
Step 1: Find the midpoint of the x-intercepts
Midpoint = (-1 + 7) / 2 = 6 / 2 = 3
Step 2: Determine the y-coordinate of the vertex
Given that the highest point has a y-coordinate of 4, we can substitute it into the equation:
4 = a(3 - h)^2 + k
Step 3: Determine the vertex coordinates
We can substitute the coordinates (h, k) of the vertex into the equation:
4 = a(3 - h)^2 + k
Since we have two variables, we can use the information from the x-intercepts to solve for a and h.
Step 4: Determine the value of a
Using the x-intercepts, we can substitute them into the equation and solve for a.
When x = -1, y = 0:
0 = a(-1 - h)^2 + k
When x = 7, y = 0:
0 = a(7 - h)^2 + k
Now we have two equations:
0 = a(-1 - h)^2 + k ...........(1)
0 = a(7 - h)^2 + k ............(2)
Step 5: Solve for k
Subtract equation (1) from (2) to eliminate k:
0 = a(7 - h)^2 - a(-1 - h)^2
Simplifying the equation gives:
0 = 49a - 14ah + ah^2 - a + 2ah + ah^2
Combine like terms:
0 = 2ah^2 - 12ah + 49a - a
0 = 2ah^2 - 12ah + 48a
Step 6: Simplify the equation and factor
Divide each term by 2a to simplify the equation and make factoring easier:
0 = h^2 - 6h + 24
We can rewrite this equation as:
h^2 - 6h + 24 = 0
Step 7: Solve for h
To solve for h, we can either factor the quadratic equation or use the quadratic formula. Let's use the quadratic formula:
h = (-(-6) ± √((-6)^2 - 4(1)(24))) / (2(1))
h = (6 ± √(36 - 96)) / 2
h = (6 ± √(-60)) / 2
Since we have a negative value inside the square root, the parabola does not have x-intercepts. This means the parabola is an upward-opening parabola.
Step 8: Find the value of a
Using one of the original equations, we can solve for a. Let's use equation (1):
When x = -1, y = 0:
0 = a(-1 - h)^2 + k
0 = a(-1 - 3)^2 + 4
0 = a(-4)^2 + 4
0 = a(16) + 4
0 = 16a + 4
-4 = 16a
a = -4/16
a = -1/4
Now we can substitute the values of a and h into the equation:
y = a(x - h)^2 + k
y = (-1/4)(x - 3)^2 + 4
Therefore, the standard equation of the parabola is y = (-1/4)(x - 3)^2 + 4.