To find the equation of the parabola with its axis parallel to the y-axis and passing through the points (0,0), (10,12), and (30,8), we can follow these steps:
### Step 1: Write the general equation of the parabola
Since the parabola's axis is parallel to the y-axis, its equation can be expressed in the form:
\[ y = ax^2 + bx + c \]
### Step 2: Use the first point (0,0) to find c
Substituting the coordinates of the point (0,0) into the equation:
\[ 0 = a(0)^2 + b(0) + c \]
This simplifies to:
\[ c = 0 \]
### Step 3: Substitute c into the equation
Now, the equation of the parabola simplifies to:
\[ y = ax^2 + bx \]
### Step 4: Use the second point (10,12) to create an equation
Substituting the coordinates of the point (10,12):
\[ 12 = a(10)^2 + b(10) \]
This simplifies to:
\[ 12 = 100a + 10b \]
Dividing the entire equation by 2 gives:
\[ 6 = 50a + 5b \]
This is our Equation 1.
### Step 5: Use the third point (30,8) to create another equation
Substituting the coordinates of the point (30,8):
\[ 8 = a(30)^2 + b(30) \]
This simplifies to:
\[ 8 = 900a + 30b \]
Dividing the entire equation by 2 gives:
\[ 4 = 450a + 15b \]
This is our Equation 2.
### Step 6: Solve the system of equations
Now we have two equations:
1. \( 6 = 50a + 5b \) (Equation 1)
2. \( 4 = 450a + 15b \) (Equation 2)
### Step 7: Multiply Equation 1 by 9
To eliminate \( b \), we can multiply Equation 1 by 9:
\[ 54 = 450a + 45b \]
This is our Equation 3.
### Step 8: Subtract Equation 2 from Equation 3
Now, we subtract Equation 2 from Equation 3:
\[ 54 - 4 = (450a + 45b) - (450a + 15b) \]
This simplifies to:
\[ 50 = 30b \]
Thus, we find:
\[ b = \frac{50}{30} = \frac{5}{3} \]
### Step 9: Substitute b back into Equation 1
Now substitute \( b = \frac{5}{3} \) back into Equation 1:
\[ 6 = 50a + 5\left(\frac{5}{3}\right) \]
This simplifies to:
\[ 6 = 50a + \frac{25}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 18 = 150a + 25 \]
Rearranging gives:
\[ 150a = 18 - 25 \]
\[ 150a = -7 \]
Thus, we find:
\[ a = -\frac{7}{150} \]
### Step 10: Write the final equation of the parabola
Now we have \( a \), \( b \), and \( c \):
- \( a = -\frac{7}{150} \)
- \( b = \frac{5}{3} \)
- \( c = 0 \)
Substituting these values back into the equation of the parabola gives:
\[ y = -\frac{7}{150}x^2 + \frac{5}{3}x \]
### Step 11: Rearranging the equation
To express it in standard form:
\[ 150y = -7x^2 + 250x \]
Rearranging gives:
\[ 7x^2 - 250x + 150y = 0 \]
This is the equation of the parabola.
