Skills covered: Intercepts; completing the square; domains and ranges; composite and inverse functions; graph transformations; exponential and logarithmic graphs.
Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.
These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if functions and graphs still feels shaky; if you are confident, go straight to Section C and time yourself.
Section A
Prerequisite core skills
Revise earlier learning
Question 1
[3 marks]
Find the coordinates of the points where y=x2−5x+6 meets the coordinate axes.
Question 2
[3 marks]
Express x2−6x+4 in completed-square form and state its minimum value.
Question 3
[3 marks]
Express 1+4x−x2 in completed-square form and state its maximum turning point.
Question 4
[2 marks]
State the range of f(x)=2(x+1)2−3, for x∈R.
Question 5
[2 marks]
Let A={−2,−1,0,1,2} and B={0,2,4,6}. Find A∩B and A∪B.
Question 6
[3 marks]
State the largest possible domain in R of each function:
(a)f(x)=x−4
(b)g(x)=x2−9x−2
Section B
Topic practice
Build individual techniques
Question 7
[4 marks]
Given f(x)=3x−2 and g(x)=x2+1, find
(a)f(g(x))
(b)g(f(x))
Question 8
[4 marks]
Given f(x)=4x+7, find f−1(x). Show that f(f−1(x))=x.
Question 9
[4 marks]
Express g(x)=3x2−12x+8 in the form a(x+b)2+c. Hence state the coordinates of its minimum turning point.
Question 10
[4 marks]
The diagram shows y=f(x), with a maximum turning point at (−2,4) and a minimum turning point at (2,−4). Sketch y=f(x−1)+2, indicating the coordinates of both turning points.
Question 11
[5 marks]
For the function f in Question 10, sketch y=−2f(x+1) and state the coordinates and natures of its turning points.
Question 12
[4 marks]
For f(x)=x2−4x+1, find simplified expressions for
(a)f(x+2)
(b)f(x)+2
(c)f(2x)
Question 13
[4 marks]
For y=2x−1+3, state the domain, range and equation of the horizontal asymptote. Give one point on the curve.
Question 14
[4 marks]
For y=log3(x+2), state the vertical asymptote and domain. Find the points at which y=0 and y=1.
Section C
Exam-style and challenge
Combine skills, exam-style
Question 15
[5 marks]
Given f(x)=32x−5, find f−1(x). Hence solve f(x)=f−1(x).
Question 16
[5 marks]
The equation x2+(k−2)x−k−3=0 has one root x=−1. Find k and hence the second root.
Question 17
[5 marks]
The diagram shows y=g(x). Sketch y=g(2−x)+1, labelling the turning point and all intercepts.
Question 18
[5 marks]
Let f(x)=x+1 and g(x)=2x−5. Find f(g(x)) and g(f(x)), stating the real domain of each.
Looking for the answers?
We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.
Are the answers to these worksheets available online?
No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.
Are these official SQA or Qualifications Scotland papers?
No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.
How should I use the three sections?
Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.
Can I print the worksheet?
Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.
