IB Maths AI SL AI 2.1 Functions and graphs 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

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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]

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