write the equation in standard form ofo the parabola containing the following point (0,0) (1,-3) (-1,-9)
To find the equation of the parabola in standard form using the given points (0,0), (1,-3), and (-1,-9), we can substitute these points into the standard form equation and solve for the coefficients.
The standard form of a parabola is given by the equation: y = ax^2 + bx + c
Substituting the point (0,0):
0 = a(0^2) + b(0) + c
0 = 0 + 0 + c
c = 0
Substituting the point (1,-3):
-3 = a(1^2) + b(1) + 0
-3 = a + b
Substituting the point (-1,-9):
-9 = a((-1)^2) + b(-1) + 0
-9 = a - b
We now have a system of equations:
-3 = a + b
-9 = a - b
Solving this system of equations, we can add the two equations together to eliminate the 'b' term:
-3 + (-9) = a + b + a - b
-12 = 2a
a = -6
Substituting the value of 'a' back into one of the equations to solve for 'b':
-3 = -6 + b
b = -3 + 6
b = 3
Finally, substituting the values of 'a' and 'b' back into the standard form equation:
y = -6x^2 + 3x
Therefore, the equation of the parabola in standard form is: y = -6x^2 + 3x
To write the equation of a parabola in standard form, we need to use the formula:
y = ax^2 + bx + c
where (x, y) are the coordinates of a point on the parabola and a, b, and c are constants to be determined.
We have three points for the parabola: (0,0), (1,-3), and (-1,-9).
Step 1: Substitute the coordinates of the first point (0,0) into the equation.
0 = a(0)^2 + b(0) + c
0 = c
So, we know that c = 0.
Step 2: Substitute the coordinates of the second point (1,-3) into the equation.
-3 = a(1)^2 + b(1) + 0
-3 = a + b
Step 3: Substitute the coordinates of the third point (-1,-9) into the equation.
-9 = a(-1)^2 + b(-1) + 0
-9 = a - b
Now, we have a system of two equations with two unknowns:
-3 = a + b → Equation 1
-9 = a - b → Equation 2
Step 4: Solve the system of equations.
From Equation 1, we can express b in terms of a:
b = -3 - a
Substitute this value of b into Equation 2:
-9 = a - (-3 - a)
-9 = a + 3 + a
-9 = 2a + 3
-12 = 2a
a = -6
Now substitute the value of a back into Equation 1:
-3 = -6 + b
b = 3
So, the values of a and b are -6 and 3, respectively.
Step 5: Write the final equation in standard form.
The equation of the parabola is:
y = -6x^2 + 3x
Therefore, the standard form of the parabola containing the points (0,0), (1,-3), and (-1,-9) is y = -6x^2 + 3x.
To find the equation in standard form of a parabola passing through the given points, we can use the general equation of a parabola in standard form:
y = ax^2 + bx + c
We have three points: (0,0), (1,-3), (-1,-9). We can substitute the x and y values from each point into the equation to create a system of equations. Let's solve it step by step:
1. First, substitute (0,0) into the equation:
0 = a(0)^2 + b(0) + c
Since any term multiplied by zero would be zero, we can simplify this equation to:
0 = c
2. Next, substitute (1,-3) into the equation:
-3 = a(1)^2 + b(1) + c
Simplify this equation to:
a + b + c = -3
3. Lastly, substitute (-1,-9) into the equation:
-9 = a(-1)^2 + b(-1) + c
Simplify this equation to:
a - b + c = -9
Now, we have a system of three equations:
Equation 1: 0 = c
Equation 2: a + b + c = -3
Equation 3: a - b + c = -9
To find the values of a, b, and c, we can solve this system of equations using any method (e.g., substitution, elimination, or matrices). Here, we will use the method of elimination:
1. Subtract Equation 3 from Equation 2:
(a + b + c) - (a - b + c) = -3 - (-9)
a + b + c - a + b - c = -3 + 9
2b = 6
b = 3
2. Substitute the value of b (3) into either Equation 2 or 3:
a + 3 + c = -3
a + c = -6
3. Substitute the value of c (0) into Equation 2:
a + 3 + 0 = -3
a = -6
Therefore, we have found that a = -6, b = 3, and c = 0. Now we can substitute these values back into the general equation of the parabola:
y = ax^2 + bx + c
y = -6x^2 + 3x + 0
y = -6x^2 + 3x
Hence, the equation in standard form of the parabola passing through the points (0,0), (1,-3), and (-1,-9) is y = -6x^2 + 3x.