IB Maths AA SL Topic 2: Functions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 1, no calculatorEasy6 marks

The function $f$ is defined for $x\in\mathbb{R}$ by

f(x)=-(x-2)^2+9.

(a)

State the coordinates of the vertex of the graph of $f$ and the equation of its axis of symmetry.

[2]

(b)

Determine the coordinates of the $x$ -intercepts of the graph of $f$

[2]

(c)

Find the $y$ -intercept of the graph of $f$ and state the maximum value of $f$

[2]

Question 2

Paper 1, no calculatorMedium6 marks

The function $f$ is defined by

f(x)=(x-2)^2+1,\qquad x\geq 2.

(a)

Explain why $f$ has an inverse function.

[1]

(b)

Find $f^{-1}(x)$ stating its domain.

[3]

(c)

Using your answer to (b), determine $(f^{-1}\circ f)(5)$ showing the value of $f(5)$

[2]

Question 3

Paper 1, no calculatorMedium7 marks

For real $k$ let $q_k(x)=x^2-4x+k$ where $x\in\mathbb R$

(a)

Solve $q_3(x)=0$

[2]

(b)

Hence solve $q_3(x)\leq 0$

[2]

(c)

Determine the values of $k$ for which $q_k(x)=0$ has: (i) two distinct real roots; (ii) two equal real roots; (iii) no real roots.

[3]

Question 4

Paper 1, no calculatorMedium5 marks

The graph of the function $r$ is shown in Figure 1. It is known that

r(x)=\frac{ax+b}{x+d}, \qquad x\ne-d,

where $a,b,d\in\mathbb R$

(a)

Determine the values of $a$ $b$ and $d$

[3]

(b)

Hence, show that $r$ is self-inverse, stating the domain of $r^{-1}$

[2]

Question 5

Paper 1, no calculatorEasy6 marks

The function $f$ is defined by $f(x)=x^2$

(a)

The graph $y=u(x)$ is obtained by translating the graph of $y=f(x)$ one unit upwards. Write down $u(x)$

[1]

(b)

The graph $y=v(x)$ is obtained by applying a vertical stretch of scale factor $2$ to the graph of $y=f(x)$ Write down $v(x)$

[1]

(c)

For parts (i) and (ii), the transformations are a translation one unit upwards and a vertical stretch of scale factor $2$

(i)

The graph $y=w(x)$ is obtained by translating the graph of $y=f(x)$ one unit upwards, then applying the vertical stretch. Determine $w(x)$

[2]

(ii)

The graph $y=z(x)$ is obtained by applying the vertical stretch first, then translating the graph of $y=f(x)$ one unit upwards. Write down $z(x)$ and state whether the graphs $y=w(x)$ and $y=z(x)$ are identical, giving a reason.

[2]

Question 6

Paper 1, no calculatorEasy4 marks

The graph of $y=f(x)$ is transformed to the graph of $y=g(x)$ where

f(x)=x^2,\qquad g(x)=-2f(x-1)+3.

(a)

State the horizontal translation used.

[1]

(b)

State the vertical stretch and reflection used to obtain $y=-2f(x-1)$ from $y=f(x-1)$

[2]

(c)

State the final vertical translation.

[1]

Question 7

Paper 1, no calculatorEasy6 marks

The graph of $f$ is shown in Figure 1. The function $g$ is defined by

g(x)=3f(x)+2.

(a)

Use Figure 1 to write down $f(1)$

[1]

(b)

State the transformations that map the graph of $f$ to the graph of $g$ in the order in which they are applied.

[2]

(c)

Hence, sketch the graph of $g$ on the same axes, marking the images of the point on $f$ for which $x=1$ and the vertex of $f$

[3]

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