GCSE Maths · Functions and Advanced Graphs · Higher · Summary

GCSE Maths Functions and Advanced Graphs (Higher): summary and key words

Part of the GCSE Maths functions and advanced graphs chapter (Higher). Chapter overview · whole chapter on one page

Chapter summary

Skill	Method
Function notation	f(a): replace every x by a (brackets). Solve f(x) = k by putting the rule in and solving the equation.
Inverse function	y = f(x); make x the subject; swap y and x. f⁻¹(b) = a when f(a) = b. Graph: reflect in y = x.
Composite function	fg(x) = f(g(x)): g first. Replace x in f by g(x). (fg)⁻¹ = g⁻¹f⁻¹.
Graph transformations	f(x) + a: up; f(x + a): a left; −f(x): reflect in x-axis; f(−x): reflect in y-axis; af(x), f(ax): stretches.
Trigonometric graphs	sin and cos: period 360°, between −1 and 1; tan: period 180°, asymptotes at 90°, 270°. Solve using the inverse key, symmetry and the period.
Exponential graphs	y = k^x passes through (0, 1), is above the x-axis. k > 1 growth; 0 < k < 1 decay.
Growth and decay	Amount = start × (multiplier)ⁿ. Multiplier 1 + r/100 or 1 − r/100. Use a table to find a time.
Circle and tangent	x² + y² = r². Tangent at (a, b): gradient −a/b (perpendicular to the radius); y − b = m(x − a).
Gradient of a curve	Draw a tangent; rise ÷ run using two points on it (instantaneous rate of change). Chord gradient = average rate of change.
Area under a graph	Area = vertical × horizontal (units multiply). Speed–time: area = distance. Estimate curves with equal strips (trapezia or rectangles).
Iteration	x_n+1 = g(x_n) with the Ans key. Show a root with a change of sign at the ends of the rounding interval.
Iterative processes	u_n+1 = au_n + b; the limit L satisfies L = aL + b.

I can use function notation to evaluate f(a) and solve f(x) = k, including with algebraic inputs.
I can find an inverse function using algebra or a function machine, and I know its graph is a reflection in y = x.
I can find composite functions such as fg(x) and solve equations involving them.
I can sketch y = f(x) + a, f(x + a), −f(x) and f(−x) from the graph of y = f(x), and identify these transformations.
I can sketch and describe the graphs of sin, cos and tan, and solve equations such as sin x = 0.3 for a range of angles.
I can sketch exponential graphs and solve growth and decay problems with multipliers.
I can write down the equation of a circle with centre the origin and find the equation of a tangent at a point.
I can estimate the gradient of a curve using a tangent and explain average and instantaneous rates of change.
I can find and estimate areas under graphs and say what the area represents.
I can use an iteration formula, show that a solution lies in an interval, and give a root to a stated accuracy.

Function: A rule that gives exactly one output for each input.
Inverse function: The function that undoes f, written f⁻¹.
Composite function: Two functions joined so the output of one is the input of the other, e.g. fg(x) = f(g(x)).
Domain: The set of allowed inputs of a function.
Transformation: A change to a graph such as a translation, reflection or stretch.
Period: The length of one complete cycle of a repeating graph.
Amplitude: Half the distance between the maximum and minimum of a wave.
Asymptote: A line that a graph gets closer and closer to but never touches.
Exponential function: A function with the unknown in the power, such as y = 2^x.
Multiplier: The number you multiply by to apply a percentage change in one step.
Tangent: A straight line that touches a curve (or circle) at exactly one point.
Chord: A straight line joining two points on a curve.
Instantaneous rate of change: The rate of change at one point: the gradient of the tangent.
Average rate of change: The change in the output divided by the change in the input: the gradient of a chord.
Iteration: Repeating a calculation, using each answer as the next input, to approach a solution.
Converge: To get closer and closer to one value.