Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions.
x-intercepts −2 and 8, highest point has y-coordinate 5
To find the standard equation of a parabola with a vertical axis, we can use the vertex form of the equation, which is given as:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the vertex is the highest point with a y-coordinate of 5.
Step 1: Find the x-coordinate of the vertex
To find the x-coordinate of the vertex, we can calculate the average of the two x-intercepts:
x-coordinate of vertex = (x-intercept 1 + x-intercept 2) / 2
= (-2 + 8) / 2
= 6 / 2
= 3
Step 2: Find the value of 'a' in the equation
To find the value of 'a', we can substitute the vertex coordinates (3, 5) into the equation:
5 = a(3 - 3)^2 + k
5 = a(0)^2 + k
5 = 0 + k
k = 5
Step 3: Write the equation in standard form
Now that we have the values of 'a', 'h', and 'k', we can write the equation of the parabola in standard form:
y = a(x - h)^2 + k
y = a(x - 3)^2 + 5
Step 4: Find the value of 'a'
To find the value of 'a', we can use one of the known points (x-intercept and y-coordinate of the highest point). Let's use the x-intercept (-2, 0):
0 = a(-2 - 3)^2 + 5
0 = a(-5)^2 + 5
0 = 25a + 5
25a = -5
a = -5/25
a = -1/5
Step 5: Substitute the value of 'a' back into the equation
Substituting the value of 'a' back into the equation, we get the final standard equation of the parabola:
y = (-1/5)(x - 3)^2 + 5
Therefore, the standard equation of the parabola with a vertical axis, x-intercepts at -2 and 8, and the highest point with a y-coordinate of 5 is y = (-1/5)(x - 3)^2 + 5.
To find the standard equation of a parabola with a vertical axis, we can use the vertex form:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the x-intercepts are -2 and 8, and the highest point has a y-coordinate of 5.
Step 1: Find the vertex of the parabola.
The x-coordinate of the vertex is the average of the x-intercepts:
h = (x-intercept1 + x-intercept2) / 2
= (-2 + 8) / 2
= 6 / 2
= 3
The y-coordinate of the vertex is the y-coordinate of the highest point:
k = 5
So the vertex is (3, 5).
Step 2: Find the value of a.
To find the value of a, we can substitute the vertex coordinates into the vertex form equation:
5 = a(3 - 3)^2 + 5
5 = a(0) + 5
5 = 5a
Dividing both sides by 5:
a = 1
Step 3: Write the standard equation.
Using the values we found, the standard equation of the parabola is:
y = 1(x - 3)^2 + 5
Therefore, the standard equation of the parabola is y = (x - 3)^2 + 5.