Exam-Style Questions on Quadratic Graph

1.	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 \(f\).

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 \(k\).

Worked Solution

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2.	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.

Worked Solution

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