Write the equation for the parabola that has x− intercepts (−2,0) and (1.2,0) and y− intercept (0,−4)
y = a (x - h)^2 + k ... (h,k) is the vertex
h is midway between the x-intercepts (roots) ... (-2 + 1.2) / 2 = -.4
from (-2,0) ... 0 = a (-2 - -.4)^2 + k ... 0 = 2.56 a + k
from (0,-4) ... -4 = a (0 - -.4)^2 + k ... -4 = .16 a + k
solve the system for a and k
subtracting equations (to eliminate k) ... 4 = 2.4 a ... a = 5/3
substituting ... -4 = (4/25 * 5/3) + k = 4/15 k ... k = -64/15
y = 5/3 (x + 2/5)^2 - 64/15
To find the equation of the parabola with given x-intercepts and y-intercept, we can start by applying the fact that the parabola passes through the points (-2, 0) and (1.2, 0).
Using the information of the x-intercepts, we know that the factors of the equation will be (x + 2) and (x - 1.2). The equation so far is (x + 2)(x - 1.2) = 0.
To find the equation of the parabola, we need to determine the leading coefficient. We can use the y-intercept (0, -4) for this purpose.
Substituting x = 0 and y = -4 into the equation, we get:
(0 + 2)(0 - 1.2) = -4
Simplifying, we have:
2(-1.2) = -4
-2.4 = -4
To make the equation (-2.4 = -4) true, we need to multiply both sides of the equation by a constant. In this case, we multiply by -1.2.
-1.2*(-2.4) = -1.2*(-4)
2.88 = 4
Now, we have the final equation of the parabola:
y = -1.2(x + 2)(x - 1.2)
Therefore, the equation for the parabola is y = -1.2(x + 2)(x - 1.2).
To find the equation of a parabola, we can start by using the vertex form equation:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex. In this case, we know that the y-intercept is at (0, -4), which means the vertex form of the equation is:
y = a(x - h)^2 - 4
Next, let's find the value of 'a'. We have two x-intercepts: (-2, 0) and (1.2, 0). When the x-coordinate is -2 or 1.2, the value of y is 0. Substituting these values into the equation, we get:
0 = a(-2 - h)^2 - 4
0 = a(1.2 - h)^2 - 4
Simplifying these equations further, we have:
4 = a(h^2 - 4h + 4)
4 = a(h^2 - 2.4h + 1.44)
Now, we can solve for 'a'. Let's consider the first equation:
4 = a(h^2 - 4h + 4)
Dividing both sides of the equation by (h^2 - 4h + 4):
4 / (h^2 - 4h + 4) = a
Similarly, for the second equation, we get:
4 / (h^2 - 2.4h + 1.44) = a
Since both equations give us the same value for 'a', we can set them equal to each other:
(h^2 - 4h + 4) = (h^2 - 2.4h + 1.44)
Expanding both sides and simplifying, we have:
h^2 - 4h + 4 = h^2 - 2.4h + 1.44
Simplifying further, we get:
1.6h = 2.56
Dividing both sides of the equation by 1.6:
h = 1.6
Now, substituting this value of h into either of the two equations, we can solve for 'a'. Let's substitute it into the first equation:
4 = a(1.6^2 - 4(1.6) + 4)
Simplifying, we find:
4 = 1.44a
Dividing both sides by 1.44, we get:
a = 2.778
Finally, we can write the equation of the parabola:
y = 2.778(x - 1.6)^2 - 4
x− intercepts (−2,0) and (1.2,0) means that
y = a(x+2)(5x-6)
Now plug in (0,-4) to find a:
a(2)(-6) = -4
-12a = -4
a = 1/3
y = 1/3 (x+2)(5x-6) = 1/3 (5x^2 + 4x - 12)