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
y = a(x+2)(x-8)
The vertex is midway between the roots, at x=3, so
y(3) = 5
a(5)(-5) = 5
a = -1/5
y = -1/5 (x+2)(x-8)
See
http://www.wolframalpha.com/input/?i=-1%2F5+%28x%2B2%29%28x-8%29
Oh, I love parabolas! They're so curvy and unpredictable! Alright, let's find the equation for this mischievous parabola.
Given that the parabola has x-intercepts at -2 and 8, we know that the parabola is symmetrical around the midpoint of these x-intercepts. The midpoint can be found by taking the average of the x-coordinates, so (8 + -2) / 2 = 6 / 2 = 3. So the vertex of the parabola is at (3, 5) because it has the highest point with a y-coordinate of 5.
Now, let's whip out our equation-writing skills. The standard equation for a parabola with a vertical axis is:
(x - h)^2 = 4p(y - k)
Where (h, k) is the coordinate of the vertex, and p is the distance between the vertex and the focus or between the vertex and the directrix. In this case, since the parabola opens upward, p is positive.
So plugging in our vertex coordinates, we have:
(x - 3)^2 = 4p(y - 5)
To find the value of p, we can use the x-intercepts. The distance between the vertex and either of the x-intercepts is called the focal width and is equal to 4p. In this case, the focal width is 8 (-2 to 8), so 8 = 4p. Solving for p, we get p = 2.
Now we can substitute the value of p into our equation:
(x - 3)^2 = 4(2)(y - 5)
Simplifying further, we have:
(x - 3)^2 = 8(y - 5)
And voila! The standard equation of the parabola is:
(x - 3)^2 = 8(y - 5)
Hope this helps brighten your day with some parabolic fun!
To find the standard equation of a parabola with a vertical axis and given conditions, we need to find the vertex and the coefficient of the squared term.
Step 1: Finding the x-coordinate of the vertex
The x-coordinate of the vertex can be found by taking the average of the x-intercepts.
x-coordinate of the vertex = (x-intercept 1 + x-intercept 2) / 2
x-coordinate of the vertex = (-2 + 8) / 2
x-coordinate of the vertex = 6 / 2
x-coordinate of the vertex = 3
Step 2: Finding the y-coordinate of the vertex
The y-coordinate of the vertex is given as 5.
Step 3: Finding the coefficient of the squared term
Since the parabola opens either upward or downward and has a vertical axis, the standard equation takes the form:
y = a(x - h)^2 + k, where (h, k) is the vertex.
So, substituting the vertex coordinates into the equation, we get:
y = a(x - 3)^2 + 5
Step 4: Finding the coefficient "a"
To find "a," we need an additional point on the parabola. We can use any point on the parabola, such as the highest point mentioned in the given conditions, which has a y-coordinate of 5.
Substitute the coordinates of the highest point into the equation:
5 = a(8 - 3)^2 + 5
5 = a(5)^2 + 5
5 = 25a + 5
25a = 0
a = 0/25
a = 0
Step 5: Writing the final standard equation
Now, substitute the value of "a" into the equation we found earlier:
y = 0(x - 3)^2 + 5
y = 5
Therefore, the standard equation of the parabola is:
y = 5.
To find the standard equation of a parabola with a vertical axis, we need to determine the values of the vertex and the coefficient "a" in the equation.
1. Determine the x-coordinate of the vertex:
The x-coordinate of the vertex can be found by taking the average of the x-intercepts.
x-coordinate of the vertex = (x-intercept 1 + x-intercept 2) / 2 = (-2 + 8) / 2 = 6 / 2 = 3.
2. Determine the y-coordinate of the vertex:
The y-coordinate of the vertex is the maximum or minimum point of the parabola (depending on whether "a" is positive or negative).
In this case, the highest point has a y-coordinate of 5.
3. Determine the value of "a":
Since the parabola has a vertical axis, the standard form of its equation is y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex. Substituting the vertex coordinates (3, 5) into the equation gives us:
5 = a(3-3)^2 + 5,
5 = a(0)^2 + 5,
5 = a(0) + 5,
5 = 0 + 5,
5 = 5.
Since 5 = 5 is always true, the value of "a" can be any real number.
4. Write the equation of the parabola:
Using the values obtained, we can write the standard equation of the parabola:
y = a(x-h)^2 + k,
y = a(x-3)^2 + 5.
Therefore, the standard equation of the parabola that satisfies the given conditions is y = a(x-3)^2 + 5, where "a" can be any real number.