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Find the equation of a parabola with horizontal axis, vertex on y axis, and passing through the points (2, 4) and (8, -2).
Solution:
To illustrate the problem, let's plot all the given items and sketch the parabola in the rectangular coordinate system as follows
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| Photo by Math Principles in Everyday Life |
As you can see in the figure, we can have two equations of parabola based on the given items in the problem. Since the axis of the parabola is horizontal and it opens to the right, the equation of a parabola in standard form is
The vertex of a parabola is located in y-axis, the coordinates of the vertex is (0, k). The above equation becomes
If (2, 4) is one of the points of a parabola, substitute the values of x and y to the above equation
If (8, -2) is one of the points of a parabola, substitute the values of x and y to the above equation
Equate
Multiply both sides of the equation by 32 and solve for the value of k
Divide both sides of the equation by 3
Equate each factor to zero and solve for the value of k
If
Solve for the value of a, we have
Therefore, the equation of a parabola in standard form is
If
Solve for the value of a, we have
Therefore, the equation of a parabola in standard form is
