Scottish Higher Higher Mathematics Syllabus

Higher Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Higher Mathematics in Scottish Higher, not a summary of it.

1. 1

   Algebra and Functions

   4 topics

   • Functions and Graphs
     • Domain, range and composite functions
     • Inverse functions and their graphs
     • Graph transformations: translations, reflections, scalings
     • Exponential and logarithmic graphs
   • Quadratic Theory
     • Completing the square and the discriminant
     • Conditions for real, equal and complex roots
     • Quadratic inequalities
   • Polynomials
     • Synthetic division and the remainder theorem
     • Factor theorem and finding roots
     • Determining unknown coefficients
   • Exponentials and Logarithms
     • Laws of logarithms
     • Solving exponential and logarithmic equations
     • Modelling experimental data with logarithmic graphs
2. 2

   Trigonometry

   4 topics

   • Radian Measure
     • Converting between degrees and radians
     • Exact values of trigonometric ratios
   • Trigonometric Equations
     • Solving equations over a given interval
     • Equations involving double angles
   • Addition and Double Angle Formulae
     • Compound angle identities
     • The wave function a cos x + b sin x = k cos(x + a)
   • Trigonometric Identities and Proof
3. 3

   Calculus

   4 topics

   • Differentiation
     • Differentiating polynomials, sin x and cos x
     • The chain rule
     • Rates of change and motion
   • Applications of Differentiation
     • Stationary points and their nature
     • Curve sketching
     • Optimisation problems
   • Integration
     • Integrating polynomials and trigonometric functions
     • Definite integrals
     • Integration by substitution (standard forms)
   • Applications of Integration
     • Area between a curve and the axes
     • Area between two curves
4. 4

   Geometry and Vectors

   4 topics

   • The Straight Line
     • Gradient, midpoint and distance
     • Equations of medians, altitudes and perpendicular bisectors
     • Parallel and perpendicular conditions
   • The Circle
     • Equation of a circle in centre-radius and general form
     • Tangents to a circle
     • Intersection of a line and a circle
   • Vectors
     • Position vectors and components in three dimensions
     • Section formula and collinearity
     • The scalar (dot) product and angle between vectors
   • Sequences and Recurrence Relations
     • Linear recurrence relations
     • Limits of sequences and the steady state

Higher Mathematics flashcards for Scottish Higher

19 of 55 cards from the Higher Mathematics deck — real questions with worked answers.

1. What does it mean for a relation to be a function?

   A function maps each element of the domain to exactly one element of the codomain; no input has more than one output.

2. State the rule for composing two functions $f$ and $g$ to form $f(g(x))$.

   $(f \circ g)(x) = f(g(x))$: apply $g$ first, then apply $f$ to the result. The domain requires $g(x)$ to lie in the domain of $f$.

3. How is the graph of $y = f(x) + a$ related to $y = f(x)$, and how is $y = f(x + a)$ related?

   $y = f(x) + a$ shifts the graph up by $a$ (vertical translation). $y = f(x + a)$ shifts it left by $a$ (horizontal translation).

4. Describe the transformations giving $y = -f(x)$ and $y = f(-x)$.

   $y = -f(x)$ reflects the graph in the $x$-axis; $y = f(-x)$ reflects it in the $y$-axis.

5. What is the relationship between the graph of a function $f$ and its inverse $f^{-1}$?

   The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$. Inputs and outputs are swapped.

6. State the discriminant and what each case tells you about the roots of $ax^{2}+bx+c=0$.

   $\Delta = b^{2} - 4ac$. If $\Delta > 0$ there are two distinct real roots; if $\Delta = 0$ a repeated (equal) real root; if $\Delta < 0$ no real roots.

7. State the quadratic formula for the roots of $ax^{2}+bx+c=0$.

   $$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$

8. Express $x^{2}+bx+c$ in completed-square form.

   $$x^{2}+bx+c = \left(x+\tfrac{b}{2}\right)^{2} + c - \frac{b^{2}}{4}$$ The turning point is at $x = -\tfrac{b}{2}$.

9. What condition on the discriminant means a line is a tangent to a parabola?

   Substituting the line into the curve gives a quadratic with $\Delta = b^{2} - 4ac = 0$ (equal roots), meaning the line meets the parabola at exactly one point.

10. State the Remainder Theorem.

    When a polynomial $p(x)$ is divided by $(x - a)$, the remainder equals $p(a)$.

11. State the Factor Theorem.

    $(x - a)$ is a factor of polynomial $p(x)$ if and only if $p(a) = 0$.

12. What does synthetic division (the nested/box method) compute for a polynomial?

    It divides a polynomial by $(x - a)$, giving the coefficients of the quotient and the remainder $p(a)$ in the final cell.

13. State the laws of logarithms for $\log_a(xy)$, $\log_a\!\left(\tfrac{x}{y}\right)$ and $\log_a(x^{n})$.

    $\log_a(xy)=\log_a x+\log_a y$; $\log_a\!\left(\tfrac{x}{y}\right)=\log_a x-\log_a y$; $\log_a(x^{n})=n\log_a x$.

14. What is the relationship between $y = a^{x}$ and $\log_a y$?

    $y = a^{x} \iff \log_a y = x$. The logarithm is the inverse of the exponential with the same base.

15. What are the values of $\log_a 1$ and $\log_a a$?

    $\log_a 1 = 0$ and $\log_a a = 1$ for any valid base $a$.

16. How do you linearise the relationship $y = a x^{n}$ using logarithms?

    Take logs: $\log y = \log a + n\log x$. A graph of $\log y$ against $\log x$ is a straight line with gradient $n$ and intercept $\log a$.

17. How do you linearise the exponential relationship $y = a b^{x}$ using logarithms?

    Take logs: $\log y = \log a + x\log b$. A graph of $\log y$ against $x$ is a straight line with gradient $\log b$ and intercept $\log a$.

18. Convert between degrees and radians: how many radians is $180^{\circ}$?

    $180^{\circ} = \pi$ radians. To convert degrees to radians multiply by $\tfrac{\pi}{180}$; for radians to degrees multiply by $\tfrac{180}{\pi}$.

19. Give the exact values of $\sin$, $\cos$ and $\tan$ at $\tfrac{\pi}{6}$ (30°).

    $\sin\tfrac{\pi}{6}=\tfrac{1}{2}$, $\cos\tfrac{\pi}{6}=\tfrac{\sqrt{3}}{2}$, $\tan\tfrac{\pi}{6}=\tfrac{1}{\sqrt{3}}$.

See more Higher Mathematics flashcards →

Planning Higher Mathematics for Scottish Higher

Higher Mathematics (Scottish Higher) FAQ