Write an equation for a parabola with x-intercepts (-2,0) and (4,0) which passes through the point (1,-18)
To find the equation for a parabola, we can use the vertex form, which is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
In this case, since the parabola passes through the points (-2,0) and (4,0), the vertex lies on the axis of symmetry, which is the line x = (−2 + 4)/2 = 1. Therefore, the vertex has x-coordinate 1.
Since the vertex (1, k) has a y-coordinate of -18, we can substitute these values into the vertex form equation:
-18 = a(1 - 1)^2 + k
-18 = a(0) + k
-18 = 0 + k
k = -18
Now that we know the values of h and k, we can substitute them into the vertex form equation:
y = a(x - 1)^2 - 18
To find the value of a, we can substitute the point (4,0) into the equation:
0 = a(4 - 1)^2 - 18
0 = a(3)^2 - 18
0 = 9a - 18
9a = 18
a = 18/9
a = 2
Therefore, the equation for the parabola is:
y = 2(x - 1)^2 - 18
To find the equation for a parabola given its x-intercepts and a point it passes through, we can use the vertex form of the equation: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Step 1: Find the vertex (h, k)
Since the parabola passes through the point (1, -18), the x-coordinate of the vertex, h, is 1. Let's substitute this value into the equation and solve for k.
-18 = a(1 - h)^2 + k
-18 = a(1 - 1)^2 + k
-18 = 0 + k
k = -18
Therefore, the vertex is (1, -18).
Step 2: Find the value of a
To find a, substitute one of the x-intercepts into the equation and solve for a.
Using the x-intercept (-2, 0):
0 = a(-2 - 1)^2 - 18
0 = a(-3)^2 - 18
0 = 9a - 18
9a = 18
a = 2
Step 3: Write the equation using the vertex form
Now that we know the values of a, h, and k, we can write the equation for the parabola.
y = a(x - h)^2 + k
y = 2(x - 1)^2 - 18
Therefore, the equation for the parabola is y = 2(x - 1)^2 - 18.