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Exam-Style Questions on Quadratic Graph

Problems on Quadratic Graph adapted from questions set in previous Mathematics exams.


IB Analysis and Approaches

Consider the function \(f(x)=\frac{1}{2}\left(2x-3\right)\left(x+5\right)\) for \(x \in \mathbb R\). The following diagram shows part of the graph of \(f\).

Quadratic Graph

For the graph of \(f\)

(a) find the coordinates of the x-intercepts.

(b) find the coordinates of the vertex.

The function \(f\) can be written in the form \(f(x) = (x+h)^2 + k\)

(c) Write down the value of \(h\) and the value of \(k\).


IB Analysis and Approaches

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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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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