The equation of the parabola passes through the parabola (1,1) and (2,4)

Options

A. `y^2=x`
B. `x^2=y`
C. `x^2=4y`
D. `y^2=4x`

Answer

To find the equation of the parabola that passes through the points (1, 1) and (2, 4), we can start by assuming a general form of the parabola. The standard form of a parabola that opens upwards is given by:

\[ y = ax^2 + bx + c \]

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

We will substitute the coordinates of the points (1, 1) and (2, 4) into the general equation to form a system of equations.

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

2. For the point (2, 4):
\[ 4 = a(2)^2 + b(2) + c \implies 4 = 4a + 2b + c \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

Now we have two equations:
1. \( a + b + c = 1 \) (Equation 1)
2. \( 4a + 2b + c = 4 \) (Equation 2)

We can eliminate \( c \) by subtracting Equation 1 from Equation 2:
\[ (4a + 2b + c) - (a + b + c) = 4 - 1 \]
This simplifies to:
\[ 3a + b = 3 \quad \text{(Equation 3)} \]

### Step 3: Express \( b \) in terms of \( a \)

From Equation 3, we can express \( b \):
\[ b = 3 - 3a \]

### Step 4: Substitute back to find \( c \)

Now substitute \( b \) back into Equation 1:
\[ a + (3 - 3a) + c = 1 \]
This simplifies to:
\[ -2a + 3 + c = 1 \]
\[ c = 1 + 2a - 3 = 2a - 2 \quad \text{(Equation 4)} \]

### Step 5: Choose a value for \( a \) to find a specific parabola

Since we have two variables \( a \) and \( c \) in terms of each other, we can choose a value for \( a \). Let's choose \( a = 1 \):
\[ b = 3 - 3(1) = 0 \]
\[ c = 2(1) - 2 = 0 \]

### Step 6: Write the final equation of the parabola

Now substituting \( a \), \( b \), and \( c \) back into the general form:
\[ y = 1x^2 + 0x + 0 \implies y = x^2 \]

Thus, the equation of the parabola is:
\[ y = x^2 \]

### Conclusion

The parabola that passes through the points (1, 1) and (2, 4) is given by the equation:
\[ y = x^2 \]