# Exam-Style Question on Quadratic Graph

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 558. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is not allowed.

The graphs of the functions $$f(x)$$, a parabola, and $$g(x)$$, a straight line, meet at exactly one point.

$$f(x) = px^2 - px$$ $$g(x) = px-5$$

where $$x \in \mathbf R \text{ and } p \in \mathbf R$$

(a) Show that $$p = 5$$

The function $$f$$ can be expressed in the form $$f(x) = 5(x-m)(x-n) \text{, where } m,n \in \mathbf R$$

(b) Find the value of $$m$$ and the value of $$n$$.

The function $$f$$ can also be expressed in the form $$f(x) = 5(x-h)^2 + k, \text{ where } h,k \in \mathbf R$$

(c) Find the value of $$h$$ and the value of $$k$$.

(d) Hence find the values of $$x$$ where the graph of $$f$$ is both negative and decreasing.

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