SATMathematics Practice Test 1

Question : 50 of 58

Marks: +1, -0

Which of the following is an equivalent form of the equation of the graph shown in the xy-plane above, from which the coordinates of vertex A can be identified as constants in the equation?

$y=(x+3)(x-5)$
$y=(x-3)(x+5)$
$y=x(x-2)-5$
$y=(x-1)^2-16$

Validate

Solution:

Any quadratic function q can be written in the form

q(x)=a(x−h)2+k

,where a, h, and k are constants and (h, k) is the vertex of the parabola when q is graphed in the coordinate plane.
(Depending on the sign of a, the constant k must be the minimum ormaximum value of q, and h is the value of x for which a

(x − h)2=0

and q(x) has value k.)
This form can be reached by completing the square in the expression that defines q.
The given equation is

y = x^2−2x−15

, and since the coefficient of x is −2,the equation can be written in terms of

(x−1)2=x2−2x+1

as follows:

y = x^2−2x−15=(x2−2x+1)−16=(x−1)2−16

.
From this form of the equation, the coefficients of the vertex can be read as

(1,−16)

Choices A and C are incorrect because the coordinates of the vertex A donot appear as constants in these equations.
Choice B is incorrect because it is not equivalent to the given equation.

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