In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions: Vertex (0,0) passing through (2,3) and axis is along x-axis.
Since the vertex is at (0,0) and the axis is along the x-axis, the equation of the parabola is of the form y² = 4ax or y² = -4ax.
The parabola passes through the point (2,3). Substituting these coordinates in the equation y² = 4ax:
Substitution: 3² = 4a(2)
Calculation: 9 = 8a
Solving for a: a = 9/8
Therefore, the equation of the parabola is:
y² = 4(9/8)x
y² = (9/2)x or 2y² = 9x
Explanation
This question tests understanding of standard parabola equations. From Example 6 in the textbook, when the focus lies on the x-axis (meaning the axis is along x-axis), the equation takes the form y² = 4ax or y² = -4ax. Since the point (2,3) has positive coordinates and lies in the first quadrant, the parabola opens to the right, so we use y² = 4ax. Substituting the point (2,3) gives us the value of a = 9/8.
Solution Steps
Step 1: Identify that vertex (0,0) with axis along x-axis means equation is y² = 4ax or y² = -4ax
Step 2: Since point (2,3) is in first quadrant, parabola opens rightward, so use y² = 4ax
Step 3: Substitute (2,3): 3² = 4a(2), giving 9 = 8a
Step 4: Solve: a = 9/8
Step 5: Write final equation: y² = 4(9/8)x = (9/2)x or 2y² = 9x