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Chapter 6: Problem 86

Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) \(y=a(x-h)^{2}+k, \quad a \neq 0\) (b) \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) (c) \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

Short Answer

Expert verified
Equation (a) is the vertex form for a parabola, equation (b) is the standard form for a parabola opening upward or downward, and equation (c) is the standard form for a parabola opening left or right. Equations (a) and (b) can be equivalent depending upon the values of 'a' and 'p'.

Step by step solution

01

Identifying Equation (a)

Equation (A): \(y=a(x-h)^{2}+k, \, a \neq 0\) is in the vertex form of a parabola \(y=a(x-h)^2+k\) where (h,k) is the vertex of the parabola and a is a non-zero constant that determines the direction and narrowness of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
02

Identifying Equation (b)

Equation (B): \((x-h)^{2}=4p(y-k), \, p \neq 0\) is the standard form of a parabola opening upward or downward. The vertex of this parabola is also (h,k), and 'p' is the distance from the vertex to the focus and from the vertex to the directrix.
03

Equating equation (a) and (b)

Equation (a) and (b) can be considered equivalent looking at their slopes and vertices. If we rewrite equation (a) as \(y-k=a(x-h)^2\), we can clearly see that when the value of \(4p\) is substituted as \(a\) in equation (a), it would take the form of equation (b). Thus they can represent the same parabolic motion, but this would depend on the values of the constants 'a' and 'p', which define the shape and orientation of the parabola.
04

Identifying Equation (c)

Equation (C): \((y-k)^{2}=4p(x-h), \, p \neq 0\) is in the standard form of a parabola that opens to the right if 'p' is positive, and to the left if 'p' is negative. Its vertex is also at (h,k).

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