Locus of Vertex

Geometry Level pending

Consider the family of parabolas whose equations have the form $y = x^2 - 2px + 3p^2 - 5p,$ where $p$ is a real number. The loci of the vetices of these parabolas satisfy the equation $y = f(x)$ . Then what is $f(7) ?$

49

71

56

63

1 solution

Tom Engelsman
Nov 8, 2020

The family of parabolas can be written in the standard form:

$y = x^2 - 2px + 3p^2 -5p = (x-p)^2 + (2p^2-5p)$

whose vertices $(p,2p^2-5p)$ lie on the curve $y = f(x) = 2x^2-5x$ . Thus, $f(7) = 2(7^2) - 5(7) = \boxed{63}.$

Locus of Vertex

1 solution

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