The value of n for which the considered quadratic equation p(x) and k(x) are such that p(x) = k(x) + n is n = -6
What is vertex form of a quadratic equation?
If a quadratic equation is written in the form
[tex]y=a(x-h)^2 + k[/tex]
then it is called to be in vertex form. It is called so because when you plot this equation's graph, you will see vertex point(peak point) is on (h,k)
This graph goes through three points, (-3,3), (3,3), and (0,-6) (the last point is the vertex of the quadratic equation).
Let the function p(x) be:
[tex]p(x) = a(x-h)^2 + k[/tex]
Since the vertex lies on the point (h,k) = (0,-6), putting h = 0, and k = -6, we get:
[tex]p(x) = ax^2 -6[/tex]
Now, as this graph passes through (x,y) = (-3,3), putting x= -3, and y (output, or say p(x)) = 3, we get:
[tex]3 = a(-3)^2 - 6\\3 = 9a-6\\9a = 9\\a =1[/tex]
Thus, the function p(x) is:
[tex]p(x) = x^2 - 6[/tex]
But since [tex]k(x) = x^2[/tex], therefore,
[tex]p(x) = x^2 - 6 = k(x) - 6[/tex]
On comparison with p(x) = k(x) + n, we deduce that n = -6
Thus, the value of n for which the considered quadratic equation p(x) and k(x) are such that p(x) = k(x) + n is n = -6
Learn more about vertex form of a quadratic equation here:
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