# Exam-Style Questions on Quadratic Graph

## Problems 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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