IB Maths AI 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 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 1Easy6 marks

The upper edge of an entrance arch is modelled by $y=-0.4(x-5)^2+10$ where $x$ is the horizontal distance in metres from the left end of the arch and $y$ is the height in metres. The model is valid for $0\leq x\leq10$

Use graphing technology for each part. Give coordinates in metres.

(a)

State the coordinates of the maximum point of the graph.

[2]

(b)

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

[2]

(c)

A horizontal crossbar is placed at a height of $6$ metres. Determine the coordinates where it meets the arch. Give your answers to two decimal places.

[2]

Question 2

Paper 2Medium15 marks

A surveyor records the elevations of points on a mountain access route. In Figure 1, $x$ is the horizontal distance, in metres, from a reference point and $y$ is the elevation, in metres.

The existing straight road runs from $A$ to $B$ A proposed extension runs from $B$ to $C$ Point $E$ is the proposed starting point of a second access track, which is to be parallel to $AB$

The maximum acceptable gradient for a road is $18\%$ where gradient is the rise divided by the horizontal distance, expressed as a percentage.

(a)

Calculate the gradient of $AB$ as a percentage.

[2]

(b)

Show that the equation of $AB$ is

y=\frac{237}{1375}x+718.4.

[2]

(c)

Hence, find the horizontal distance from $A$ at which $AB$ reaches an elevation of $750$ m. Give your answer to the nearest metre.

[2]

(d)

The second access track passes through $E$ and is parallel to $AB$

Write its equation first in the form $y-y_1=m(x-x_1)$ then in the form $y=mx+c$ Hence, find its elevation when $x=400$ m. Give your answer to the nearest $0.1$ m.

[6]

(e)

Calculate the gradient of $BC$ as a percentage and determine whether the proposed extension meets the maximum acceptable gradient.

[3]

Question 3

Paper 1Hard8 marks

An urban greenhouse produces and sells seedlings in complete batches of 1000. Let $q$ be the number of batches produced in a year, where $q$ is an integer and $0\leq q\leq80$

The selling price per seedling is modelled by

p(q)=96-1.2q

dollars. The annual operating cost is modelled by

C(q)=0.4q^2+12q+500

thousand dollars. The greenhouse also pays an annual permit fee of $F$ thousand dollars. Assume that all seedlings produced are sold.

Let $P_F(q)$ denote the annual profit, in thousands of dollars. The greenhouse considers a year acceptable if its profit is at least $350 000.

(a)

Write an expression for $P_F(q)$

[1]

(b)

When $F=200$ use your graphic display calculator to determine all allowed values of $q$ for which the year is acceptable.

[2]

(c)

Determine the greatest annual permit fee that would still allow the greenhouse to have an acceptable year, and the corresponding production.

[4]

(d)

State one limitation of this model when it is used to make the decision in part (c).

[1]

Question 4

Paper 1Easy7 marks

A technician records the volume, $V$ of water in a container as it is filled from a tap.

$t$ (minutes)	0	2	4	6
$V$ (litres)	2.6	4.0	5.5	7.0

(a)

State one feature of the measurements that supports using a linear model for $V$

[1]

(b)

Using the measurements at $t=0$ and $t=6$ determine the gradient $m$ and intercept $c$ of the linear model $V(t)=mt+c$ Hence write down the model.

[3]

(c)

Use your model to estimate the volume at $t=5$ minutes, giving your answer to the nearest $0.1$ litre. State whether this prediction involves extrapolation, giving a reason.

[3]

Question 5

Paper 1Medium14 marks

A horticulture centre monitors the depth of water in a shallow pond during a controlled fill-and-drain test. Let $t$ be the time, in hours, since the test began, and let $h$ be the depth of water, in centimetres.

$t$	0	2	5	7
$h$	1.3	7.6	10.1	7.0

(a)

Choose the most suitable model from a linear, quadratic or exponential model for these data. Justify your choice.

[2]

(b)

Using only the observations at $t=0$ $t=2$ and $t=5$ fit a quadratic model of the form $h=at^2+bt+c$ Show that

h=-0.463t^2+4.08t+1.30,

where the coefficients are given to 3 significant figures. Retain unrounded values for later parts.

[3]

(c)

Use your fitted model to estimate the depth at $t=7$ Compare your estimate with the recorded depth.

[2]

(d)

Use your fitted model to estimate the time at which the depth is $8$ cm on the falling part of the curve. Give your answer to the nearest $0.01$ hour.

[3]

(e)

State a reasonable domain for using this model, based on the observations. Use the model to estimate the depth at $t=10$ and comment on whether this prediction is reasonable.

[4]

Question 6

Paper 1Medium14 marks

An engineer records the water height, $h$ cm, in a reservoir while a pump fills it. The pump remains on throughout a planned 8-minute run, and there is no outlet. The measurements shown in Figure 1 were taken during the first 5 minutes. Heights are recorded to the nearest $0.1$ cm.

(a)

Using the shape of the data in Figure 1, choose a linear or quadratic model for $h$ in terms of $t$ Justify your choice.

[2]

(b)

Let $h=at^2+bt+c$ Using the measurements at $t=0$ $t=2$ and $t=4$ show that a fitted model is

h=-0.5375t^2+6.975t+3.8.

[4]

(c)

Hence, using this model, determine the first time during the 5-minute measurement interval at which the model predicts $h=25.0$ cm. Give your answer to the nearest $0.01$ minute.

[3]

(d)

At $t=3$ compare the model’s predicted height with the recorded height. State what this comparison suggests about the model’s fit at that time.

[2]

(e)

Use the fitted model to predict the height after 8 minutes. State whether this prediction should be considered reliable, giving two reasons.

[3]

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