Graphs Study Guide

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer exponents of the variables. For example, quadratic functions like y = ax² + bx + c are polynomials of degree 2, and cubic functions like y = ax³ + bx² + cx + d are polynomials of degree 3. Higher methods for these functions often involve understanding their roots (where the function crosses the x-axis), turning points (local maxima or minima), and long-term behaviour (as x approaches positive or negative infinity).

Reciprocal functions are of the form y = k/x, where k is a constant. More generally, they can be expressed as y = k/P(x), where P(x) is a polynomial. Key features of reciprocal functions include asymptotes – lines that the graph approaches but never touches. For y = k/x, there are vertical asymptotes at x=0 and horizontal asymptotes at y=0. For functions like y = 1/(x-a) + b, the vertical asymptote is at x=a and the horizontal asymptote at y=b. Understanding these asymptotes is crucial for sketching and analysing the graphs of reciprocal functions. As x approaches a value that makes the denominator zero, the function's magnitude tends towards infinity. As x approaches positive or negative infinity, the function tends towards the horizontal asymptote.

Higher methods involve using algebraic techniques to find roots of cubic or quartic polynomials (e.g., factor theorem, algebraic division), analysing the behaviour of functions using differentiation to find turning points (though this is A-level content and not required at GCSE), and sketching graphs based on their roots, asymptotes, and general shape. For reciprocal functions, understanding transformations (translations, reflections, stretches) applied to the basic y=1/x graph is also important. For example, y = 1/(x+2) is a translation of y = 1/x by 2 units to the left.

Exponential functions are mathematical functions of the form y = ab^x, where 'a' is a non-zero constant, 'b' is a positive real number not equal to 1, and 'x' is the independent variable, a real number. In this function, the variable 'x' appears in the exponent. These functions model situations where a quantity increases or decreases at a rate proportional to its current value, meaning growth or decay is multiplicative rather than additive. Examples include population growth, compound interest, radioactive decay, and cooling of objects. The constant 'a' represents the initial value of y when x = 0, because any non-zero number raised to the power of 0 is 1 (ab^0 = a * 1 = a). The constant 'b' is the growth or decay factor; if b > 1, it represents growth, and if 0 < b < 1, it represents decay. For example, if b = 1.5, the quantity increases by 50% for each unit increase in x. If b = 0.8, the quantity decreases by 20% for each unit increase in x. Solving problems involving exponential functions often requires using logarithms to find the exponent 'x' when other values are known. Graphically, exponential growth functions increase rapidly, while exponential decay functions decrease rapidly towards zero but never actually reach it, approaching the x-axis as an asymptote. Developing and applying higher methods for these functions involves manipulating the equations, solving for unknown parameters, and interpreting their behaviour in real-world contexts.