IB Maths AI HL 2.1 Functions Sl Content Questions

Practise IB Mathematics AI HL 2.1 by analysing advanced function models, transformations, inverse and regression evidence with technology.

Exam points

Represent functions with correct notation, domain, range and inverse relationships, evaluating and composing them in context.
Construct, sketch and transform function graphs, identifying intercepts, extrema, symmetry, asymptotes and intersections.
Select and use function models, interpret parameters and use technology to fit, solve or predict within a valid domain.
Use graph and model features to answer thresholds, optimisation and applied questions with appropriate units.
Check a function or model using substitution, graph behaviour, data fit, assumptions and contextual reasonableness.

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—Modelling with functions SL 2.6—Modelling process

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Study 2.1 Functions - SL content conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 5]

Line $L_{1}$ is tangent to the graph of a function f(x) at the point P(3,-1). Line $L_{2}$ is given by the equation $y=-\frac{1}{2} x-\frac{5}{2}$ and is perpendicular to $L_{1}$ .

(a)

Write down the gradient of $L_{1}$ .

[ 1 ]

(b)

Find the equation of $L_{1}$ in the form y=m x+c.

[ 2 ]

attempt to substitute their part (a) and point (3,-1) into the slope-intercept form or point-slope form of an equation
$-1=2 \times 3+c \quad$ OR $\quad y+1=2(x-3)$
y=2 x-7
Note: Equation must be in the form y=m x+c for A1 to be awarded.

(c)

Show that $L_{2}$ is not the line that is normal to f(x) at point P .

[ 2 ]

METHOD 1
attempt to show that P does not lie on $L_{2}$
e.g. $-\frac{1}{2}(3)-\frac{5}{2} \quad \mathbf{OR}$ graph showing $L_{2}$ and P in approximate correct locations
$-1 \neq-\frac{1}{2}(3)-\frac{5}{2}(-1 \neq-4) \mathbf{OR}(3,-1)$ does not lie on the graph of $L_{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 2
attempt to find the equation of the normal line at (3,-1)
$\left(-1=-\frac{1}{2}(3)+c \quad\right.$ OR $\left.\quad y+1=-\frac{1}{2}(x-3)\right)$
the normal line is $y=-\frac{1}{2} x+\frac{1}{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 3
attempt to find the intersection of $L_{1}$ and $L_{2}$

Intersection of y=2 x-7 and $y=-\frac{1}{2} x-\frac{5}{2}$ is (1.8,-3.4)
$x=1.8 \neq 3 \quad$ OR $\quad y=-3.4 \neq-1$
hence $L_{2}$ is not the normal line to f(x) at point P

Note: Accept equivalent written arguments provided values are seen.
Methods 1 and 2 are independent of the answers in (a) and (b) but FT marks can be given for Method 3.

[Maximum number: 2]

The function f is defined by $f(x)=4 \ln (2 x-3)$ , where $x>\frac{3}{2}$ .
The graph of y=f(x) is obtained from the graph of $y=\ln x$ by a sequence of three transformations.

The graph of y=g(x) is shown.
The graph passes through the points (-2,-2),(-1,0),(0,2),(1,3),(2,5) and (3,6).

Find $(f \circ g)(2)$ .

$\quad g(2)=5$ $f(5)=7.78 \quad(=7.78364 \ldots, 4 \ln 7)$

[Maximum number: 3]

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

k=A \mathrm{e}^{-\frac{c}{T}}

This equation links a variable k with the temperature T, where A and c are positive constants and T>0.

Given that $\lim_{T\to\infty}k=A$ and $\lim_{T\to0}k=0$ , sketch the graph of k against T.

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Covered syllabus points

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Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question 3