Write the equation in standard form of the parabola containg the following point (0,1) (1,-3) (-1,-9)
set up a system of three equations
To find the standard form of the equation of a parabola, we can use the general equation:
y = ax^2 + bx + c
We can substitute the given points into this equation to set up a system of three equations.
1) For the point (0,1):
1 = a(0)^2 + b(0) + c
1 = 0 + 0 + c
c = 1
So, our equation becomes:
y = ax^2 + bx + 1
2) For the point (1,-3):
-3 = a(1)^2 + b(1) + 1
-3 = a + b + 1
3) For the point (-1,-9):
-9 = a(-1)^2 + b(-1) + 1
-9 = a - b + 1
Now we can solve this system of equations to find the values of a and b.
From equation (2):
a + b = -4 -> a = -4 - b
Substituting this value of a into equation (3):
-9 = (-4 - b) - b + 1
-9 = -8 - 2b
2b = -1
b = -1/2
Now we can substitute the value of b back into equation (2):
a - 1/2 = -4
a = -4 + 1/2
a = -7/2
Therefore, the equation of the parabola in standard form is:
y = (-7/2)x^2 - (1/2)x + 1
To find the equation in standard form of a parabola containing the given points (0, 1), (1, -3), and (-1, -9), you can use the general form of a parabolic equation, which is y = ax^2 + bx + c.
Step 1: Substitute the x and y coordinates of the first point (0, 1) into the equation.
1 = a(0)^2 + b(0) + c
1 = 0 + 0 + c
c = 1
Step 2: Substitute the x and y coordinates of the second point (1, -3) into the equation.
-3 = a(1)^2 + b(1) + c
-3 = a + b + 1
a + b = -4 (Equation 1)
Step 3: Substitute the x and y coordinates of the third point (-1, -9) into the equation.
-9 = a(-1)^2 + b(-1) + c
-9 = a - b + 1
a - b = -10 (Equation 2)
In standard form, we like to express the equation as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. So, we need to rewrite Equation 1 and Equation 2 to fit this form.
Step 4: Multiply both sides of Equation 1 by x.
(x)(a + b) = -4x
Step 5: Multiply both sides of Equation 2 by x.
(x)(a - b) = -10x
Step 6: Expand both sides of the new equations.
ax + bx = -4x
ax - bx = -10x
Step 7: Simplify and combine like terms.
2ax = -4x
2ax + 10x = 0
2ax + 10x + 0 = 0
Step 8: Factor out the x terms.
2x(a + 5) = 0
Step 9: Set each factor equal to zero and solve for a and b.
2x = 0 or (a + 5) = 0
If 2x = 0, then x = 0.
If (a + 5) = 0, then a = -5.
So, we have the values x = 0 and a = -5.
Step 10: Substitute the value of a into Equation 2.
-5 - b = -10
b = -10 + 5
b = -5
Therefore, the equation in standard form of the parabola is:
y = -5x^2 - 5x + 1
To find the equation of the parabola in standard form, we need to find the quadratic equation that represents it. A quadratic equation can be written as:
y = ax^2 + bx + c
To determine the values of a, b, and c, we can substitute the given points (0, 1), (1, -3), and (-1, -9) into the equation and solve the resulting system of equations.
Let's start by substituting the point (0, 1) into the equation:
1 = a(0)^2 + b(0) + c
1 = c
So we have c = 1.
Now let's substitute the point (1, -3) into the equation:
-3 = a(1)^2 + b(1) + 1
-3 = a + b + 1
Next, let's substitute the point (-1, -9) into the equation:
-9 = a(-1)^2 + b(-1) + 1
-9 = a - b + 1
Now we have a system of three equations:
1 = c (equation 1)
-3 = a + b + 1 (equation 2)
-9 = a - b + 1 (equation 3)
To solve this system, we can use any algebraic method such as substitution or elimination. Let's use the elimination method.
Subtracting equation 3 from equation 2 eliminates the 'a' term:
(-3) - (-9) = (a + b + 1) - (a - b + 1)
6 = 2b
Dividing by 2, we get b = 3.
Now substituting this value of b into equation 2 gives us:
-3 = a + 3 + 1
-3 = a + 4
Subtracting 4 from both sides, we get a = -7.
So the values of a, b, and c are a = -7, b = 3, and c = 1.
Now we can write the equation in standard form by substituting these values back into the equation:
y = ax^2 + bx + c
y = -7x^2 + 3x + 1
Therefore, the equation in standard form of the parabola containing the points (0, 1), (1, -3), and (-1, -9) is y = -7x^2 + 3x + 1.