Functions & Graphs · Guided Practice

Functions & Graphs

12 questions with answers and video solutions. Try each one before revealing the answer.

1
Identify the vertical asymptotes of the curve defined by the equation: y=x2+1x2−x−6\displaystyle y=\frac{x^2+1}{x^2-x-6}
Video solution coming soon

No hints on this one — its full written solution is with the answer.

2
Identify the vertical asymptote of the curve defined by the equation: y=sin⁡xx(x−1)\displaystyle y=\frac{\sin x}{x(x-1)}
Video solution coming soon

No hints on this one — its full written solution is with the answer.

3
A function is defined on a suitable domain by f(x)=3x2+2x2−2\displaystyle f(x)=\frac{3x^2+2}{x^2-2}.
Obtain equations for the asymptotes of the graph of y=f(x)\displaystyle y=f(x).
Video solution coming soon

No hints on this one — its full written solution is with the answer.

4
A function is defined on a suitable domain by f(x)=x3−xx2−2x−8\displaystyle f(x)=\frac{x^3-x}{x^2-2x-8}.
Obtain equations for the asymptotes of the graph of y=f(x)\displaystyle y=f(x).
Video solution coming soon

No hints on this one — its full written solution is with the answer.

5
The function f\displaystyle f is defined on a suitable domain by f(x)=x2+n\displaystyle f(x)=x^2+n, where the constant n∈R\displaystyle n\in\mathbb{R}.
State whether f\displaystyle f is odd, even or neither.
Give a reason for your answer.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

6
The function f\displaystyle f is defined on a suitable domain by f(x)=x3cos⁡x\displaystyle f(x)=x^3\cos x.
State whether f\displaystyle f is odd, even or neither.
Give a reason for your answer.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

7
The function f\displaystyle f is defined on a suitable domain by f(x)=e2x\displaystyle f(x)=e^{2x}.
State whether f\displaystyle f is odd, even or neither.
Give a reason for your answer.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

8
A function is defined on a suitable domain by f(x)=3x2+2x2−2\displaystyle f(x)=\frac{3x^2+2}{x^2-2}.
Determine whether the graph of y=f(x)\displaystyle y=f(x) has any points of inflection.
Justify your answer.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

9
Determine the coordinates and natures of all stationary points and points of inflection on the graph of y=2x3−12x2−30x+9\displaystyle y=2x^3-12x^2-30x+9.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

10
2016 Exemplar Q10
Find the coordinates of the point of inflexion on the graph of y=sin⁡x+tan⁡x\displaystyle y=\sin x+\tan x, where −π2<x<π2\displaystyle -\frac{\pi}{2}< x<\frac{\pi}{2}.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

11
2019 Q3
(1, 1)2 Marks

The function f(x)\displaystyle f(x) is defined by f(x)=x2−a2.\displaystyle f(x) = x^2 - a^2. The graph of y=f(x)\displaystyle y = f(x) is shown in the diagram.

Graph of y = x^2 - a^2

(a)State whether f(x)\displaystyle f(x) is odd, even or neither. Give a reason for your answer.

(b)Sketch the graph of y=∣f(x)∣.\displaystyle y = |f(x)|.

Video solution coming soon
12
2024 P1 Q5
(2, 2)4 Marks

The function f(x)\displaystyle f(x) is defined by f(x)=x3−x\displaystyle f(x) = x^3 - x, x∈R.\displaystyle x \in \mathbb{R}.

(a)Determine whether f(x)\displaystyle f(x) is even, odd or neither.

(b)Show that the graph of y=f(x)\displaystyle y = f(x) has a point of inflection.

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More functions & graphs questions

New questions, modelled on past paper questions and checked against their marking instructions. Not reprints — you have not seen these before.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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