What is the equation, in standard form, of a parabola that contains the following points?
Left parenthesis negative 2 comma negative 20 right parenthesis, left parenthesis 0 comma negative 4 right parenthesis, left parenthesis 4 comma negative 20 right parenthesis
(1 point)
Responses
y equals negative 2.5 x squared plus 5 x
Image with alt text: y equals negative 2.5 x squared plus 5 x
y equals negative x squared plus 4 x minus 4
Image with alt text: y equals negative x squared plus 4 x minus 4
y equals negative 2 x squared plus 4 x minus 4
Image with alt text: y equals negative 2 x squared plus 4 x minus 4
y equals negative 2.25 x squared plus 4.5 x minus 2
The correct equation, in standard form, of the parabola is:
y = -2.25x^2 + 4.5x - 2
To determine the equation of a parabola that contains the given points, we can use the standard form of the equation: y = ax^2 + bx + c.
First, let's substitute the coordinates of the first point (-2, -20) into the equation: -20 = a(-2)^2 + b(-2) + c.
Next, substitute the coordinates of the second point (0, -4): -4 = a(0)^2 + b(0) + c.
Lastly, substitute the coordinates of the third point (4, -20): -20 = a(4)^2 + b(4) + c.
Now we have a system of three equations with three variables (a, b, and c). We can solve this system to find the values of a, b, and c, which will give us the equation of the parabola.
Solving the system of equations, we find that a = -2.5, b = 5, and c = 0.
Therefore, the equation of the parabola in standard form is y = -2.5x^2 + 5x.
So, the correct answer is:
y = -2.5x^2 + 5x.
To find the equation of a parabola in standard form that passes through the given points (-2, -20), (0, -4), and (4, -20), we can use the vertex form of a parabola equation: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
First, let's find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:
h = -(sum of x-coordinates of the given points) / (number of points)
h = -((-2 + 0 + 4) / 3)
h = -2/3
Next, we find the y-coordinate of the vertex. We substitute the x-coordinate of the vertex into one of the given points. Let's use (0, -4):
-4 = a(0 - (-2/3))^2 + k
-4 = a(2/3)^2 + k
-4 = (4/9)a + k
Now, let's substitute the x and y coordinates of another given point into the equation to find a. Let's use (4, -20):
-20 = a(4 - (-2/3))^2 + k
-20 = a(14/3)^2 + k
We now have two equations:
-4 = (4/9)a + k
-20 = (196/9)a + k
We can solve these equations simultaneously to find the values of a and k.
Solving the first equation for k, we get:
k = -4 - (4/9)a
Substituting this expression for k into the second equation, we have:
-20 = (196/9)a + (-4 - (4/9)a)
-20 = (196/9)a - (36/9)a - (4/9)a
-20 = (156/9)a
a = -20 * (9/156)
a = -5/39
Substituting the value of a back into the first equation to find k, we have:
-4 = (4/9)(-5/39) + k
-4 = (-20/351) + k
k = -4 + (20/351)
k = (-1408/351 + 20/351)
k = -1388/351
k = -4
Therefore, the equation of the parabola in standard form is:
y = (-5/39)(x + 2/3)^2 - 4