To find the equation of a parabola, we can generally express it in the form:
\[ y = ax^2 + bx + c \]
Given the points:
- \((-1, -7)\)
- \((1, 13)\)
- \((3, -7)\)
- \((5, -67)\)
We will use three of these points to create a system of equations.
Step 1: Set up equations using three points
-
For the point \((-1, -7)\): \[ -7 = a(-1)^2 + b(-1) + c \implies -7 = a - b + c \quad \text{(Equation 1)} \]
-
For the point \((1, 13)\): \[ 13 = a(1)^2 + b(1) + c \implies 13 = a + b + c \quad \text{(Equation 2)} \]
-
For the point \((3, -7)\): \[ -7 = a(3)^2 + b(3) + c \implies -7 = 9a + 3b + c \quad \text{(Equation 3)} \]
Step 2: Solve the system of equations
Now we have the following system:
- \( a - b + c = -7 \)
- \( a + b + c = 13 \)
- \( 9a + 3b + c = -7 \)
We can solve this system step by step.
Step 3: Eliminate \(c\)
Subtract Equation 1 from Equation 2:
\[ (a + b + c) - (a - b + c) = 13 - (-7) \] \[ 2b = 20 \implies b = 10 \]
Now substitute \(b = 10\) into Equations 1 and 2 to find \(a\) and \(c\).
Using Equation 1:
\[ a - 10 + c = -7 \implies a + c = 3 \quad \text{(Equation 4)} \]
Using Equation 2:
\[ a + 10 + c = 13 \implies a + c = 3 \quad \text{(Confirmed)} \]
Now substitute \(b = 10\) into Equation 3:
\[ 9a + 3(10) + c = -7 \implies 9a + 30 + c = -7 \] \[ 9a + c = -37 \quad \text{(Equation 5)} \]
Step 4: Solve Equations 4 and 5
Now we have:
- \( a + c = 3 \) (Equation 4)
- \( 9a + c = -37 \) (Equation 5)
Subtract Equation 4 from Equation 5:
\[ (9a + c) - (a + c) = -37 - 3 \] \[ 8a = -40 \implies a = -5 \]
Now substitute \(a = -5\) back into Equation 4 to find \(c\):
\[ -5 + c = 3 \implies c = 8 \]
Summary of coefficients
Now we have:
- \(a = -5\)
- \(b = 10\)
- \(c = 8\)
So, the equation of the parabola is:
\[ y = -5x^2 + 10x + 8 \]
Final Answer
The equation of the parabola is:
\[ y = -5x^2 + 10x + 8 \]
1 answer