Pure Mathematics Practice: Graphing Techniques

Original practice material

These four structured questions total 34 marks. They include assessed sketching and algebraic interpretation suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Graphing Techniques Answers.

Question 1: Transformations

The graph of $y=f(x)$ is transformed to $y=2f(x-3)-1$.

(a) Describe the transformations in a valid order. [3]

(b) If $(1,4)$ lies on $y=f(x)$, find the corresponding point on the transformed graph. [2]

(c) If $f$ has range $[-2,5]$, find the range of $2f(x-3)-1$. [2]

[Total: 7]

Question 2: Rational Graph

Let

$$y=\frac{3x-1}{x+2}.$$

(a) Write the function in the form $a+\frac{b}{x+2}$. [2]

(b) State the vertical and horizontal asymptotes. [2]

(c) Find the intercepts with the coordinate axes. [3]

(d) Sketch the graph, showing the asymptotes, intercepts and the correct branch positions. [3]

[Total: 10]

Question 3: Modulus Graph

Let

$$g(x)=x^2-4x+3.$$

(a) Find the roots and vertex of $y=g(x)$. [3]

(b) Sketch $y=|g(x)|$, labelling its intercepts and any non-differentiable points. [3]

(c) Hence, or otherwise, solve $|g(x)|=3$. [4]

[Total: 10]

Question 4: Symmetry, Range and an Inequality

Consider

$$h(x)=5-(x-2{)}^2,-1\le x\le 5.$$

(a) State the axis of symmetry. [1]

(b) Find the range of $h$. [3]

(c) Solve $h(x)\ge 1$ on the stated domain. [3]

[Total: 7]