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Exam-Style Questions on Quadratic GraphProblems on Quadratic Graph adapted from questions set in previous Mathematics exams. |
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 \(k\).
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.
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