SQA Higher Mathematics Area 1 Expressions and Functions: the straight line, functions, trigonometry, logs and vectors

What Area 1 actually demands

Expressions and Functions is the algebra and geometry toolkit of Higher Mathematics. The examiners test fluent manipulation, exact non-calculator work, and the ability to move between an algebraic rule and the graph it produces. This guide walks through all five topics of the area, then sets out the patterns the SQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

The straight line

The area opens with the straight line. The gradient between two points is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, and it equals $\tan\theta$ where $\theta$ is the angle the line makes with the positive x-axis. A line through $(a, b)$ with gradient $m$ has equation $y - b = m(x - a)$. Parallel lines share a gradient; perpendicular lines have gradients whose product is $-1$. These tools give the medians, altitudes and perpendicular bisectors of a triangle from coordinates.

Functions and graphs

Functions and graphs covers composite functions $f(g(x))$ (apply $g$ first), inverse functions whose graph reflects $y = f(x)$ in the line $y = x$, and the standard transformations: $f(x) + a$ shifts up, $f(x + a)$ shifts left, $-f(x)$ and $f(-x)$ reflect, and $kf(x)$ and $f(kx)$ stretch. The exponential graph and its inverse the logarithmic graph appear here too.

Trigonometry and radians

Trigonometry and radians introduces radian measure, where $\pi$ radians equal $180^\circ$, and the exact values from the standard triangles. For a graph $y = a\sin(bx + c)$ the amplitude is $|a|$, the period is $\dfrac{2\pi}{b}$, and the phase shift comes from $c$. Because Paper 1 is non-calculator, exact surd and fraction values are essential.

Exponentials and logarithms

Exponentials and logarithms uses the laws $\log_a x + \log_a y = \log_a(xy)$, $\log_a x - \log_a y = \log_a\!\left(\dfrac{x}{y}\right)$ and $\log_a(x^n) = n\log_a x$ to simplify expressions and solve equations with the unknown in the power. Logarithms also straighten experimental data: $y = kx^n$ and $y = ab^x$ become straight lines on log axes, giving the constants from the gradient and intercept.

Vectors

Vectors works in three dimensions. The magnitude of $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is $\sqrt{a^2 + b^2 + c^2}$, a unit vector is the vector divided by its magnitude, and the scalar product $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$ both gives the angle through $\cos\theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$ and tests perpendicularity, since perpendicular vectors have scalar product zero.

How Area 1 is examined

A typical SQA profile for Expressions and Functions:

• Exact non-calculator work. Paper 1 rewards surd, fraction and exact trigonometric answers done by hand.
• Linking algebra to graphs. Transformations, inverse graphs and the shapes of exponential and log curves.
• Coordinate geometry. Gradients, perpendicular lines and the special lines of a triangle.
• Applied vectors. Angles and perpendicularity in three dimensions using the scalar product.

Check your knowledge

A mix of recall and method questions covering Area 1. Attempt them, then check against the solutions.

1. Write the equation of the line through $(2, 5)$ with gradient $3$. (2 marks)
2. Given $f(x) = 2x$ and $g(x) = x + 1$, find $f(g(x))$. (2 marks)
3. State the period of $y = \sin(3x)$ in radians. (1 mark)
4. Simplify $\log_2 16 + \log_2 4$. (2 marks)
5. Find the magnitude of $\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$. (2 marks)