Write an equation for a parabola with x-intercepts (-1,0) and (3,0) which passes through the point (1,-16)
To find an equation for the given parabola, we can start by using the intercept form of a parabola equation: y = a(x - h)(x - k), where (h, k) is the vertex of the parabola.
Given that the parabola has the x-intercepts (-1,0) and (3,0), we know that the factors (x - h) are (x + 1) and (x - 3).
So far, our equation is y = a(x + 1)(x - 3).
We also know that the parabola passes through the point (1, -16). Plugging in the x and y coordinates of this point into the equation, we get:
-16 = a(1 + 1)(1 - 3)
-16 = a(2)(-2)
-16 = a(-4)
-16 = -4a
Solving for a, we divide both sides of the equation by -4:
a = -16 / -4
a = 4
Now that we have the value for a, we can substitute it into the quadratic equation:
y = 4(x + 1)(x - 3)
Thus, the equation for the given parabola is y = 4(x + 1)(x - 3).
To find an equation for the parabola, we can first determine the vertex, and then use the vertex form of the equation for a parabola.
Step 1: Find the x-coordinate of the vertex.
The x-coordinate of the vertex can be found by taking the average of the x-intercepts.
Vertex_x = (x_intercept1 + x_intercept2)/2
= (-1 + 3)/2
= 2/2
= 1
Step 2: Find the y-coordinate of the vertex.
Since the parabola passes through the point (1, -16), we can substitute these values into the equation to find the y-coordinate of the vertex.
-16 = a(1-1)^2 + b(1) + c
-16 = a + b + c (Equation 1)
Step 3: Substitute the vertex coordinates into the general form of the equation.
Using the vertex form of the equation for a parabola:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Substituting (1, -16) into the equation:
-16 = a(1 - 1)^2 + k
-16 = k
Step 4: Substitute the vertex coordinates into the equation.
Using the vertex form of the equation for a parabola:
y = a(x - h)^2 + k
Substituting (1, -16) and 1 for h and k, respectively, we get:
y = a(x - 1)^2 - 16
Step 5: Substitute one of the x-intercepts to find the value of a.
Using (-1, 0), we can substitute x = -1 and y = 0 into the equation.
0 = a(-1 - 1)^2 - 16
0 = a(-2)^2 - 16
0 = 4a - 16
16 = 4a
a = 4
Step 6: Substitute the value of a into the equation.
Substituting the value of a (a = 4) into the equation, we have:
y = 4(x - 1)^2 - 16
Therefore, the equation of the parabola with x-intercepts (-1,0) and (3,0) that passes through the point (1,-16) is:
y = 4(x - 1)^2 - 16