GMAT Math : Graphing a quadratic function

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Example Question #21 : Graphing A Quadratic Function

The parabolas of the functions f(x) $\displaystyle f(x)$ and g(x) $\displaystyle g(x)$ on the coordinate plane have the same vertex.

If we define $f(x) = \frac{4}{5}(x-9)^{2}+ 6$ $\displaystyle f(x) = \frac{4}{5}(x-9)^{2}+ 6$, which of the following is a possible equation for g(x) $\displaystyle g(x)$ ?

Screen shot 2016 02 10 at 12.25.12 pm

Possible Answers:

$g(x) = \frac{4}{5}(x-9)^{2}-2$ $\displaystyle g(x) = \frac{4}{5}(x-9)^{2}-2$

$g(x) = \frac{8}{5}(x-18)^{2}+ 12$ $\displaystyle g(x) = \frac{8}{5}(x-18)^{2}+ 12$

None of the other responses gives a correct answer.

$g(x) = \frac{4}{5}(x-7)^{2}+ 6$ $\displaystyle g(x) = \frac{4}{5}(x-7)^{2}+ 6$

$g(x) = \frac{3}{5}(x-9)^{2}+ 6$ $\displaystyle g(x) = \frac{3}{5}(x-9)^{2}+ 6$

Correct answer:

$g(x) = \frac{3}{5}(x-9)^{2}+ 6$ $\displaystyle g(x) = \frac{3}{5}(x-9)^{2}+ 6$

Explanation:

The eqiatopm of f(x) $\displaystyle f(x)$ is given in the vertex form

$f(x) =a(x-h)^{2}+ k$ $\displaystyle f(x) =a(x-h)^{2}+ k$,

so the vertex of its parabola is (h,k ) $\displaystyle (h,k )$. The graphs of f(x) $\displaystyle f(x)$ and g(x) $\displaystyle g(x)$ are parabolas with the same vertex, so they must have the same values for $\displaystyle h$ and $\displaystyle k$.

For the function $f(x) = \frac{4}{5}(x-9)^{2}+ 6$ $\displaystyle f(x) = \frac{4}{5}(x-9)^{2}+ 6$, h = 9 $\displaystyle h = 9$ and k = 6 $\displaystyle k = 6$.

Screen shot 2016 02 10 at 12.25.12 pm

Of the five choices, the only equation of g(x) $\displaystyle g(x)$ that has these same values, and that therefore has a parabola with the same vertex, is $g(x) = \frac{3}{5}(x-9)^{2}+ 6$ $\displaystyle g(x) = \frac{3}{5}(x-9)^{2}+ 6$.

Screen shot 2016 02 10 at 12.27.19 pm

To verify, graph both functions on the same grid.

Screen shot 2016 02 10 at 12.28.14 pm

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GMAT Math : Graphing a quadratic function

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Example Question #21 : Graphing A Quadratic Function

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