Write an equation of a parabola with the x-intercepts at [1/2,0] and (-6,0) which passes through the point (0,6)
To find the equation of a parabola, we will use the vertex form. The vertex form of a parabola equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Since the x-intercepts are at [1/2, 0] and (-6, 0), we know that the parabola crosses the x-axis at these points. This means that the vertex lies between these two points. The x-coordinate of the vertex is the average of the x-intercepts:
h = (1/2 + (-6))/2 = -5/4
Since the parabola passes through the point (0,6), we can substitute these values into the equation:
6 = a(0 - (-5/4))^2 + k
6 = a(5/4)^2 + k
6 = a(25/16) + k
To find the value of k, we substitute the y-coordinate of the vertex (k = 0) into the equation:
0 = a(-5/4)^2 + k
0 = a(25/16) + k
0 = 25a/16 + 0
25a/16 = 0
This tells us that a = 0. Substituting this value of a into the equation, we have:
6 = 0(25/16) + k
6 = 0 + k
k = 6
Therefore, the equation of the parabola is:
y = 0(x - (-5/4))^2 + 6
y = 6
To find the equation of a parabola, we can start by using the vertex form of the equation: y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
Given the x-intercepts at (1/2, 0) and (-6, 0), we know that the parabola has two solutions when y = 0. This means that the vertex of the parabola lies midway between these two x-intercepts.
The x-coordinate of the vertex can be found by averaging the x-values of the x-intercepts:
(-6 + 1/2) / 2 = -11/4 / 2 = -11/8 = -1.375
Since the parabola passes through the point (0, 6), we can substitute these values into the equation:
6 = a(0 - (-1.375))^2 + k
Simplifying further:
6 = a(1.375)^2 + k
6 = a(1.890625) + k
6 = 1.890625a + k
Now we have two unknowns (a and k). To solve for them, let's use the fact that the parabola passes through one of the x-intercepts, (1/2, 0). We can substitute these values into the equation:
0 = a(0.5 - (-1.375))^2 + k
0 = a(1.875)^2 + k
0 = a(3.515625) + k
Since both equations equal zero, we can set the two equations equal to each other:
1.890625a + k = 3.515625a + k
By canceling out the k terms on both sides, the equation simplifies to:
1.890625a = 3.515625a
Divide both sides by a:
1.890625 = 3.515625
This equation is not true, which means that a should not exist. Thus, we have an indeterminate form.
Therefore, no parabola exists that simultaneously satisfies the given conditions.
To find the equation of a parabola, we need to determine the values of its coefficients. In this case, we have the x-intercepts at (1/2, 0) and (-6, 0), and the point (0, 6) which the parabola passes through.
First, let's start by finding the vertex of the parabola, which is the midpoint of the x-intercepts. The midpoint formula is given by:
midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2 )
Using the coordinates of the x-intercepts, we have:
midpoint = ( (1/2 + -6) / 2, (0 + 0) / 2 )
= ( -5/4, 0 )
So the vertex of the parabola is (-5/4, 0).
Next, let's use the vertex form of a parabola equation, which is given by:
y = a(x - h)^2 + k
Where (h, k) is the vertex.
Substituting the vertex coordinates, we have:
y = a(x - (-5/4))^2 + 0
y = a(x+5/4)^2
Now, we substitute the point (0, 6) into the equation to find the value of 'a':
6 = a(0 + 5/4)^2
6 = a(25/16)
a = 6 * (16/25)
a = 96/25
Finally, we substitute the value of 'a' back into the equation:
y = (96/25)(x + 5/4)^2
So, the equation of the parabola is y = (96/25)(x + 5/4)^2.