Write the equation, in standard form, of the parabola containing the following points: (0, 1), (1, -3), (-1, -9).
You must set up a system of three equations in your work in order to get full credit for this question.
(1 point)
Responses
y=3x2−5x−1
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=
3
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2
−
5
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−
1
y is equal to 3 x squared minus 5 x minus 1
y=−x2+5x+3
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=
−
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2
+
5
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+
3
y is equal to negative x squared plus 5 x plus 3
y=−7x2+3x+1
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=
−
7
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2
+
3
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+
1
y is equal to negative 7 x squared plus 3 x plus 1
y=−3x2+7x+1
To find the equation of the parabola in standard form, we can use the general equation of a parabola, which is y = ax^2 + bx + c.
We can substitute the coordinates of the given points into this equation to form a system of equations.
When x = 0, y = 1:
1 = a(0)^2 + b(0) + c
1 = 0 + 0 + c
c = 1
When x = 1, y = -3:
-3 = a(1)^2 + b(1) + 1
-3 = a + b + 1
When x = -1, y = -9:
-9 = a(-1)^2 + b(-1) + 1
-9 = a - b + 1
Now, we have a system of equations:
1 = c
-3 = a + b + 1
-9 = a - b + 1
From the first equation, we know that c = 1. We can substitute this value into the second and third equations:
-3 = a + b + 1
-9 = a - b + 1
Rearranging these equations, we get:
a + b = -4
a - b = -10
To solve this system of equations, we can add the two equations together:
(a + b) + (a - b) = -4 + (-10)
2a = -14
a = -7
Substituting this value back into one of the equations, we can solve for b:
-7 + b = -10
b = -10 + 7
b = -3
Therefore, the equation of the parabola in standard form is:
y = -7x^2 - 3x + 1
To find the equation of the parabola in standard form that contains the given points, we need to set up a system of equations.
The standard form of a quadratic equation is given by:
y = ax^2 + bx + c
We can substitute the coordinates of each point into this equation to form a system of equations.
Using the first point (0, 1):
1 = a(0)^2 + b(0) + c
1 = c
This means that the constant term, c, is equal to 1.
Using the second point (1, -3):
-3 = a(1)^2 + b(1) + c
-3 = a + b + 1
Using the third point (-1, -9):
-9 = a(-1)^2 + b(-1) + c
-9 = a - b + 1
Now, we have three equations:
1 = c (Equation 1)
-3 = a + b + 1 (Equation 2)
-9 = a - b + 1 (Equation 3)
We can solve this system of equations to find the values of a and b.
From Equation 2, we can rewrite it as:
a + b = -4
Adding Equation 3 and Equation 2:
-9 + a - b + a + b = 2
Simplifying:
2a - 8 = 2
Adding 8 to both sides:
2a = 10
Dividing both sides by 2:
a = 5
Substituting the value of a into Equation 2:
5 + b = -4
Subtracting 5 from both sides:
b = -9
Now that we have the values of a and b, we can substitute them into Equation 1 to find the value of c:
1 = c
Therefore, the equation of the parabola in standard form is:
y = 5x^2 - 9x + 1
To find the equation of the parabola containing the points (0, 1), (1, -3), and (-1, -9), we need to set up a system of equations using the general equation of a parabola in standard form: y = ax^2 + bx + c.
Substituting the given points into the equation, we get three equations:
1) At (0, 1): 1 = a(0)^2 + b(0) + c
Simplifying, we get c = 1.
2) At (1, -3): -3 = a(1)^2 + b(1) + c
Simplifying, we get a + b + c = -3.
3) At (-1, -9): -9 = a(-1)^2 + b(-1) + c
Simplifying, we get a - b + c = -9.
Now we have a system of equations:
c = 1 (equation 1)
a + b + c = -3 (equation 2)
a - b + c = -9 (equation 3)
To solve this system of equations, we can subtract equation 3 from equation 2:
(a + b + c) - (a - b + c) = -3 - (-9)
Simplifying, we get 2b = 6
Dividing both sides by 2, we get b = 3.
Substituting b = 3 into equation 2, we get:
a + 3 + 1 = -3
Simplifying, we get a = -7.
So, the values of a, b, and c are a = -7, b = 3, and c = 1.
Therefore, the equation of the parabola in standard form is:
y = -7x^2 + 3x + 1.