Write the equation for the parabola that has x− intercepts (−2,0) and (1.2,0) and y− intercept (0,−4)
since you have the roots, you know that
y = a(x+2)(x - 6/5)
since y(0) = -4,
a(2)(-6/5) = -4
a = -4 * -5/12 = 5/3
so,
y = 5/3 (x+2)(x - 6/5)
= 1/3 (x+2)(5x-6)
To find the equation of the parabola, we need to use the standard form of the quadratic equation, which is:
y = a(x - h)² + k
Where (h, k) is the vertex of the parabola.
Let's start by finding the vertex of the parabola using the x-intercepts (-2, 0) and (1.2, 0).
The formula for finding the x-coordinate of the vertex is given by:
h = (x1 + x2) / 2
h = (-2 + 1.2) / 2
h = -0.8 / 2
h = -0.4
Now we can substitute the x-coordinate of the vertex (-0.4) and the y-intercept (0, -4) into the equation to find the value of 'a':
-4 = a(-0.4 - 0)² + (-4)
-4 = a(-0.4)² - 4
-4 = a(0.16) - 4
-4 = 0.16a - 4
Now we can solve for 'a':
0.16a = -4 + 4
0.16a = 0
a = 0/0.16
a = 0
Therefore, the value of 'a' is 0.
Now we have the value of 'a' as well as the vertex (h, k) = (-0.4, -4), we can now write the equation of the parabola:
y = a(x - h)² + k
Substituting 'a' and the vertex values:
y = 0(x - (-0.4))² - 4
y = 0(x + 0.4)² - 4
Finally, simplifying the equation:
y = 0(x + 0.4)² - 4
y = 0(x² + 0.8x + 0.16) - 4
y = 0.8x + 0 - 4
y = 0.8x - 4
Therefore, the equation of the parabola is y = 0.8x - 4.