Question

Diagram below shows the curve of a quadratic function

f(x) = -x + kx-6.
A is the point of intersection of the quadratic graph and y-axis. The x-intercepts are -6 and 2.


The function can be expressed in the form

`f(x) = (1)/(2) (x - p {)}^(2) + q`
find the value of q and

How to do it step by step

Answer: q=-8, k=2

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Answer 1

q = -8, k = 2.

r = -6.

Explanation:

f(x) = (x - p)^2 + q

This is the vertex form of a quadratic where the vertex is at the point (p, q).

Now the x intercepts are at -6 and 2 and the curve is symmetrical about the line x = p.

The value of p is the midpoint of -6 and 2 which is (-6+2) / 2 = -2.

So we have:

f(x) = 1/2(x - -2)^2 + q

f(x) = 1/2(x + 2)^2 + q

Now the graph passes through the point (2, 0) , where it intersects the x axis, therefore, substituting x = 2 and f(x) = 0:

0 = 1/2(2 + 2)^2 + q

0 = 1/2*16 + q

0 = 8 + q

q = -8.

Now convert this to standard form to find k:

f(x) = 1/2(x + 2)^2 - 8

f(x) = 1/2(x^2 + 4x + 4) - 8

f(x) = 1/2x^2 + 2x + 2 - 8

f(x) = 1/2x^2 + 2x - 6

So k = 2.

The r is the y coordinate when x = 0.

so r = 1/2(0+2)^2 - 8

= -6.