Understanding the Parabola Equation with Vertex and Point
The task at hand involves determining the equation of a parabola given its vertex and a point that lies on the parabola. This involves a detailed step-by-step process that showcases how to apply the vertex form of a parabola and convert it to the standard form.
The Vertex Form of a Parabola
The general vertex form of a parabola with a vertical axis of symmetry is:
[ y a(x - h)^2 k ] Where (h) and (k) represent the coordinates of the vertex. In this problem, the vertex is given as ((2, 4)) and the parabola contains the point ((1, 1)).Step 1: Applying the Vertex Form
Substituting the vertex coordinates into the vertex form equation:
[ y a(x - 2)^2 4 ]This simplifies the equation to a given vertex and maintains the general structure for further calculations.
Step 2: Determining the Value of (a)
To find the value of (a), we use the point that lies on the parabola, which is ((1, 1)). Substituting (x 1) and (y 1) into the equation, we get:
[ 1 a(1 - 2)^2 4 ]Simplifying the equation:
[ 1 a(-1)^2 4 ] [ 1 a 4 ]Solving for (a):
[ a 1 - 4 -3 ]Therefore, the value of (a) is (-3).
Step 3: Converting to Standard Form
Now that we have (a -3), substituting it back into the vertex form equation:
[ y -3(x - 2)^2 4 ]Expanding this equation to convert it to the standard form [ax^2 bx c]:
[ y -3(x^2 - 4x 4) 4 ] [ y -3x^2 12x - 12 4 ] [ y -3x^2 12x - 8 ]The final equation of the parabola in standard form is:
[ y -3x^2 12x - 8 ]Verification
To verify the equation, we can substitute the point ((2, 4)) back into the equation:
[ 4 -3(2)^2 12(2) - 8 ] [ 4 -12 24 - 8 ] [ 4 4 ]The equation holds true, confirming the correctness of the parabola's equation.
Conclusion
Through this detailed process, we have successfully determined the equation of a parabola given its vertex and a point on the parabola. The final equation in standard form is:
[ y -3x^2 12x - 8 ]This approach can be applied to similar problems involving parabolas with given vertices and points on the curve.