The parabola whose equation is `y=ax^(2)+bx+c` passes through the points `(-3, -40), (0, 29), and (-1, 10)`. What is an equation of the line of symmetry?

Options

A. `x=(17)/(4)`
B. `x=(9)/(2)`
C. `x=5`
D. `x=6`

Answer

To find the equation of the line of symmetry for the parabola given by the equation \( y = ax^2 + bx + c \) that passes through the points \((-3, -40)\), \((0, 29)\), and \((-1, 10)\), we will follow these steps:

### Step 1: Set up the equations using the given points

We know that the points lie on the parabola, so we can substitute each point into the equation \( y = ax^2 + bx + c \) to form a system of equations.

1. For the point \((-3, -40)\):
\[ -40 = a(-3)^2 + b(-3) + c \implies -40 = 9a - 3b + c \quad \text{(Equation 1)} \]

2. For the point \((0, 29)\):
\[ 29 = a(0)^2 + b(0) + c \implies c = 29 \quad \text{(Equation 2)} \]

3. For the point \((-1, 10)\):
\[ 10 = a(-1)^2 + b(-1) + c \implies 10 = a - b + c \quad \text{(Equation 3)} \]

### Step 2: Substitute \(c\) into the equations

From Equation 2, we have \( c = 29 \). We can substitute this value into Equations 1 and 3.

- Substituting into Equation 1:
\[ -40 = 9a - 3b + 29 \implies 9a - 3b = -40 - 29 \implies 9a - 3b = -69 \]
Dividing by 3:
\[ 3a - b = -23 \quad \text{(Equation 4)} \]

- Substituting into Equation 3:
\[ 10 = a - b + 29 \implies a - b = 10 - 29 \implies a - b = -19 \quad \text{(Equation 5)} \]

### Step 3: Solve the system of equations

Now we have a system of two equations (Equations 4 and 5):
1. \( 3a - b = -23 \) (Equation 4)
2. \( a - b = -19 \) (Equation 5)

We can subtract Equation 5 from Equation 4:
\[ (3a - b) - (a - b) = -23 - (-19) \]
This simplifies to:
\[ 3a - b - a + b = -23 + 19 \implies 2a = -4 \implies a = -2 \]

### Step 4: Find \(b\)

Now substitute \( a = -2 \) back into Equation 5:
\[ -2 - b = -19 \implies -b = -19 + 2 \implies -b = -17 \implies b = 17 \]

### Step 5: Find the line of symmetry

The equation of the line of symmetry of a parabola given by \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{17}{2(-2)} = \frac{17}{4} \]

### Final Answer

The equation of the line of symmetry is:
\[ \boxed{x = \frac{17}{4}} \]