IB Maths AA HL 2.1 Functions Sl Content Questions

Practise IB Mathematics AA HL 2.1 by solving advanced function, transformation, inverse, composition and modelling problems with exact reasoning.

Exam points

Evaluate and compose functions or construct inverses while enforcing domain/range and one-to-one restrictions.
Use graph featuresâ€”roots, extrema, intercepts, asymptotes, intersections and rangesâ€”to determine parameters or solution counts.
Solve and classify quadratic or rational equations/inequalities, including discriminant, asymptotic and parameter constraints.
Analyse exponential and logarithmic graphs and models, solving for values or parameters and interpreting inverse relationships.
Apply ordered graph transformations and compositions, deriving formulae and mapping key coordinates, periods or domains.

Covered syllabus points

SL 2.1—Straight lines SL 2.2—Functions and inverses SL 2.3—Graphing functions SL 2.4—Key graph features SL 2.5—Composite and inverse functions SL 2.6—Quadratic functions SL 2.7—Quadratic equations and inequalities SL 2.8—Reciprocal and rational functions SL 2.9—Exponential and logarithmic functions SL 2.10—Solving equations SL 2.11—Graph transformations

[Maximum number: 4]

This question investigates a ratio of lengths found from the line passing through the points of inflexion of a quartic curve of the form $y=x^{4}-m x^{3}+n x$ .
The curve $y=x^{4}-3 x^{3}+3 x$ has points of inflexion at B and C . The line passing through B and C intersects the curve again at points A and D . This is shown in the following graph.

(a)

Show that the equation of the line through B and C is y=-0.375 x.

[ 2 ]

$\quad y$ -intercept =0 (as line passes through (0,0) )

Note: Award R1 for correctly substituting point B or point C to show that y=0.

Award R1 for y=-0.375 x+0.

Note: Award AOFT if their answer to (b) doesn't lead to the given answer, but condone $\frac{-0.563}{1.5}$ .

so equation is y=-0.375 x
AG
Note: Award at most A1 R0 for working backwards to verify points B and C lie on the given line.

(b)

Show that the equation of the line through B and C is $y=\left(-\frac{m^{3}}{8}+n\right) x$ .

[ 2 ]

attempt to divide their $y_{C}$ by their $x_{C}$ OR substitute into $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

\begin{aligned} & \text { e.g. }\left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \div \frac{m}{2} \\ & \left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \times \frac{2}{m} \text { OR equivalent manipulation leading to given answer } \end{aligned}

Note: Award AO FT if their answer does not lead to the AG.

=-\frac{m^{3}}{8}+n \text { and } y \text {-intercept }=0

\text { so equation is } y=\left(-\frac{m^{3}}{8}+n\right) x

Note: Award at most M1A0 for working backwards to verify points B and C lie on the given line.

[Maximum number: 1]

A function f is defined by $f(x)=\arcsin \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$ .

State the domain of $g^{-1}$ .

domain is $-\frac{\pi}{2} \leq x<\frac{\pi}{2}$
Note: Accept correct alternative notations, for example, $\left\lfloor-\frac{\pi}{2}, \frac{\pi}{2}\right\rfloor$ or $\left\lfloor-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

Marking guidance:

Accept [-1.57,1.57[ if correct to 3 s.f.

[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

Sketch the curve of y for $-10 \leq x \leq 10$ .

[ 1 ]

(b)

State the coordinates of the points where the curve crosses the x-axis.

[ 1 ]

$\quad(-4,0),(0,0),(4,0)$

Note: Award A0 if additional points are given.

[Maximum number: 1]

Sketch the curve of y for $-10 \leq x \leq 10$ .

[Maximum number: 1]

State the coordinates of the points where the curve crosses the x-axis.

$\quad(-4,0),(0,0),(4,0)$

Note: Award A0 if additional points are given.

[Maximum number: 5]

Consider the function $f(x)=a x^{3}+b x^{2}+c x+d$ , where $x \in \mathbb{R}$ and $a, b, c, d \in \mathbb{R}$ .

Sketch the graphs of y=g(x) and $y=g^{-1}(x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

Figure for Answer to Question 6 — IB Maths AA HL

Note: Award A1 for correct 'shape' of g (allow non-stationary point of inflexion)
Award A1 for each correct intercept of g
Award M1 for attempt to reflect their graph in y=x, A1 for completely correct $g^{-1}$ including intercepts

[Maximum number: 4]

A particle P moves along the x-axis. The velocity of P is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time t seconds, where $v(t)=4+4 t-3 t^{2}$ for $0 \leq t \leq 3$ . When t=0, P is at the origin O .

Sketch a graph of v against t, clearly showing any points of intersection with the axes.

valid approach to solve $4+4 t-3 t^{2}=0$ (may be seen in part (a))
(2-t)(2+3 t) OR $\frac{-4 \pm \sqrt{16+48}}{-6}$
correct x - intercept on the graph at t=2
correct domain from 0 to 3 starting at (0,4)
vertex in approximately correct place for $t=\frac{2}{3}$ and v>4

Figure for Answer to Question 7 — IB Maths AA HL

Note: The following two A marks may only be awarded if the shape is a concave down parabola. These two marks are independent of each other and the (M1).
Note: The 3 must be clearly indicated.

[Maximum number: 8]

The functions f and g are defined by

\begin{aligned} & f(x)=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \\ & g(x)=\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \end{aligned}

(a)

Let h(x)=n f(x)+g(x) where $n \in \mathbb{R}, n>1$ .

[ 8 ]

(i)

By forming a quadratic equation in $\mathrm{e}^{x}$ , solve the equation h(x)=k, where $k \in \mathbb{R}^{+}$ .

[ 5 ]

$\quad(n+1) \mathrm{e}^{2 x}-2 k \mathrm{e}^{x}+(n-1)=0$ $\mathrm{e}^{x}=\frac{2 k \pm \sqrt{4 k^{2}-4\left(n^{2}-1\right)}}{2(n+1)}$ $x=\ln \left(\frac{k \pm \sqrt{k^{2}-n^{2}+1}}{n+1}\right)$

(ii)

Hence or otherwise show that the equation h(x)=k has two real solutions provided that $k>\sqrt{n^{2}-1}$ and $k \in \mathbb{R}^{+}$ .

[ 3 ]

for two real solutions, we require $k>\sqrt{k^{2}-n^{2}+1}$
and we also require $k^{2}-n^{2}+1>0$ $k^{2}>n^{2}-1$ $\Rightarrow k>\sqrt{n^{2}-1}\left(k \in \mathbb{R}^{+}\right)$

[Maximum number: 5]

Consider the function defined by $f(x)=\frac{k x-5}{x-k}$ , where $x \in \mathbb{R} \backslash\{k\}$ and $k^{2} \neq 5$ .

(a)

State the equation of the vertical asymptote on the graph of y=f(x).

[ 1 ]

(b)

State the equation of the horizontal asymptote on the graph of y=f(x).

[ 1 ]

(c)

Sketch the graph of y=f(x), stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

[ 3 ]

attempt to draw both branches of a rectangular hyperbola M1
x=3 and y=3
A1
$\left(0, \frac{5}{3}\right)$ and $\left(\frac{5}{3}, 0\right)$

Figure for Answer to Question (c) — IB Maths AA HL

[Maximum number: 3]

Consider the function f defined by $f(x)=\ln \left(x^{2}-16\right)$ for x>4.
The following diagram shows part of the graph of f which crosses the x-axis at point A , with coordinates (a, 0). The line L is the tangent to the graph of f at the point B .

Find the exact value of a.

$\quad \ln \left(x^{2}-16\right)=0$

\begin{aligned} & \mathrm{e}^{0}=x^{2}-16(=1) \\ & x^{2}=17 \text { OR } x= \pm \sqrt{17} \end{aligned}

$a=\sqrt{17}$

[Maximum number: 3]

Consider the function f defined by $f(x)=90 \mathrm{e}^{-0.5 x}$ for $x \in \mathbb{R}^{+}$ .
The graph of f and the line y=x intersect at point P .

(a) attempt to find the point of intersection of the graph of f and the line y=x (M1)
$x=5.56619 \ldots$
=5.57

(a)

Find the x-coordinate of P .

The line L has a gradient of -1 and is a tangent to the graph of f at the point Q .

[ 2 ]

attempt to find the point of intersection of the graph of f and the line y=x (M1)
$x=5.56619 \ldots$
=5.57

(b)

Find the x-coordinate of the point where L intersects the line y=x.

[ 1 ]

$x=\ln 45+1(=4.81)$

[Maximum number: 3]

Consider the function $f(x)=a x^{3}+b x^{2}+c x+d$ , where $x \in \mathbb{R}$ and $a, b, c, d \in \mathbb{R}$ .

State each of the transformations in the order in which they are applied.

(c) translation through $\binom{2}{0}$ ,

Note: This can be seen anywhere.

EITHER
a stretch scale factor $\frac{1}{2}$ parallel to the y-axis then a translation through $\binom{0}{-4} \quad \boldsymbol{A} 2$
OR
a translation through $\binom{0}{-8}$ then a stretch scale factor $\frac{1}{2}$ parallel to the y-axis A2

Note: Accept 'shift' for translation, but do not accept 'move'. Accept 'scaling' for 'stretch'.

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)

Question 4

Question 5

Question 6

Question 7

Question 8

Question (a)

Question (i)

Question (ii)

Question 9

Question (a)

Question (b)

Question (c)

Question 10

Question 11

Question (a)

Question (b)

Question 12