Graphs Revision Notes

What is a Polynomial Function?

A polynomial function is an expression made up of terms added or subtracted, where each term is a constant multiplied by a variable raised to a non-negative integer power. For example, y = 3x^4 - 2x^2 + x - 7 is a polynomial. The highest power of the variable is called the degree of the polynomial. For instance, y = 5x + 2 is a polynomial of degree 1 (linear), y = x^2 - 4 is a polynomial of degree 2 (quadratic), and y = x^3 + 2x is a polynomial of degree 3 (cubic). Polynomial functions are always defined for all real numbers and their graphs are smooth and continuous curves with no breaks or sharp points. The general form is P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a non-negative integer and a_n ≠ 0.

What is a Reciprocal Function?

A reciprocal function is a function where the variable appears in the denominator. The most basic form is y = k/x, where k is a constant. For example, y = 1/x, y = 5/x, or y = -2/x are reciprocal functions. A key feature of these functions is that they are undefined when the denominator is equal to zero. For y = k/x, this means x cannot equal zero. This creates a vertical asymptote at x = 0 (the y-axis). Additionally, as the absolute value of x becomes very large (either positive or negative), the value of y approaches zero, creating a horizontal asymptote at y = 0 (the x-axis). The graph of a reciprocal function typically consists of two separate branches, forming a hyperbola.

Key Characteristics of Polynomial Graphs

The shape of a polynomial graph depends on its degree and the sign of its leading coefficient. Linear functions (degree 1) are straight lines. Quadratic functions (degree 2) are parabolas, opening upwards if the leading coefficient is positive and downwards if negative. Cubic functions (degree 3) can have up to two turning points and generally start and end in opposite directions – for example, from bottom-left to top-right if the leading coefficient is positive. Higher degree polynomials can have more turning points, but the maximum number of turning points is always one less than the degree (n-1). All polynomial graphs are continuous and smooth curves.

Key Characteristics of Reciprocal Graphs

The graph of a simple reciprocal function like y = k/x has two distinct parts, called branches. These branches symmetrically approach two asymptotes: a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The graph never touches or crosses these lines. If k is positive, the branches are in the first and third quadrants. If k is negative (e.g., y = -1/x), the branches are in the second and fourth quadrants. It's important to remember that the function is undefined at the vertical asymptote, meaning there's a break in the graph at that point.

What are Polynomial Functions?

Polynomial functions are expressions formed by adding and subtracting terms. Each term consists of a number (coefficient) multiplied by one or more variables raised to non-negative integer powers. Examples include linear functions (like y = 3x + 2) where the highest power of x is 1, and quadratic functions (like y = x² - 4x + 5) where the highest power of x is 2. The graphs of polynomial functions are smooth and continuous curves. Linear functions produce straight lines, and quadratic functions produce parabolas (U-shaped curves).

What are Reciprocal Functions?

Reciprocal functions have a variable in the denominator, for example, y = k/x, where 'k' is a constant. The most important feature is that the function is undefined when the denominator is zero (i.e., x ≠ 0). This creates a 'gap' or discontinuity in the graph. As x approaches zero, y approaches positive or negative infinity. As |x| increases, y approaches zero.

Graphing Reciprocal Functions

The graph of a simple reciprocal function like y = 1/x or y = -1/x consists of two separate curves called hyperbolas. For y = 1/x, these curves lie in the first and third quadrants. For y = -1/x, they lie in the second and fourth quadrants. These curves approach, but never touch, the x-axis and y-axis. These axes are called asymptotes. You should be able to recognise these characteristic shapes.

Evaluating Functions

To find the value of a polynomial or reciprocal function for a given x, substitute that value of x into the function's equation and calculate y. For example, if y = x² + 3 and x = 2, then y = (2)² + 3 = 4 + 3 = 7. If y = 10/x and x = 5, then y = 10/5 = 2. Always be careful with signs and order of operations.

Understanding Polynomial Functions

Polynomial functions are expressions of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n ≠ 0, n is a non-negative integer (the degree of the polynomial), and a_i are constants. For GCSE, you primarily encounter quadratic (degree 2) and cubic (degree 3) polynomials. Key characteristics include roots (values of x for which P(x) = 0), which can be found by factorising or using the quadratic formula for degree 2. The Factor Theorem states that if P(a) = 0, then (x-a) is a factor of P(x). This is vital for finding roots of higher-degree polynomials by testing integer factors of the constant term. The number of roots corresponds to the degree of the polynomial, potentially including repeated or non-real roots.

Graphing Polynomials: Roots and Behaviour

The roots of a polynomial function are the points where its graph intersects the x-axis. For a quadratic function, the graph is a parabola, opening upwards if the x² coefficient is positive and downwards if negative. Cubic functions have an 'S' or 'N' shape, and can have up to three real roots. The overall shape of a polynomial graph is determined by its highest degree term. For large positive x, the graph of P(x) behaves like a_n x^n. For large negative x, its behaviour depends on both a_n and whether n is even or odd. Sketching polynomials involves identifying roots, the y-intercept (P(0)), and considering the end behaviour. For example, a cubic with a positive leading coefficient starts low and ends high.

Reciprocal Functions and Asymptotes

Reciprocal functions are typically in the form y = k/x or y = k/(x-a) + b. They are characterised by asymptotes, which are lines the graph approaches but never touches. For y = k/x:

1. Vertical Asymptote: x = 0 (the y-axis). The function is undefined when the denominator is zero.
2. Horizontal Asymptote: y = 0 (the x-axis). As x becomes very large (positive or negative), y approaches 0.

For y = k/(x-a) + b:

1. Vertical Asymptote: x = a. This is because x-a = 0 when x=a.
2. Horizontal Asymptote: y = b. As x → ±∞, k/(x-a) → 0, so y → b. Understanding these asymptotes is key to sketching reciprocal graphs and solving related problems.

Transformations of Reciprocal Graphs

Transformations can be applied to the basic reciprocal function y = 1/x:

• Horizontal translation: y = 1/(x-a) shifts the graph 'a' units to the right (if a > 0) or left (if a < 0). The vertical asymptote moves to x = a.
• Vertical translation: y = 1/x + b shifts the graph 'b' units upwards (if b > 0) or downwards (if b < 0). The horizontal asymptote moves to y = b.
• Stretch/Reflection: y = k/x stretches the graph vertically by a factor of |k|. If k is negative, the graph is reflected across the x-axis. By combining these transformations, you can sketch more complex reciprocal functions like y = (2x+3)/(x-1) by first rewriting it as y = A + B/(x-1) using algebraic division or inspection.

Basic Form and Characteristics

An exponential function is generally expressed as y = ab^x. Here, 'a' represents the initial amount or the y-intercept (when x = 0), and 'b' is the growth/decay factor per unit change in x. For growth, b > 1. For decay, 0 < b < 1. The base 'b' must be positive and not equal to 1. The variable 'x' is the exponent. The domain of an exponential function is all real numbers, while the range is y > 0 (if a > 0) or y < 0 (if a < 0), as the function never reaches zero, approaching it as an asymptote. Understanding these basic characteristics is crucial for interpreting and applying exponential models.

Growth and Decay Factors

When 'b' is expressed as 1 + r, 'r' is the growth rate (as a decimal). For example, a 10% increase means r = 0.10, so b = 1.10. When 'b' is expressed as 1 - r, 'r' is the decay rate. For example, a 10% decrease means r = 0.10, so b = 0.90. It's important to remember that these rates are per unit of 'x'. If 'x' represents time in years, 'r' is the annual rate. If the rate is given over a different time period, adjustments might be needed to determine 'b' for the given unit of 'x'.

Solving Exponential Equations using Logarithms

To solve for an unknown exponent 'x' in an equation like ab^x = c, you first need to isolate the exponential term: b^x = c/a. Then, take the logarithm of both sides. For example, using the natural logarithm (ln) or logarithm base 10 (log): ln(b^x) = ln(c/a). Using the logarithm property log(M^p) = p log(M), we get x ln(b) = ln(c/a). Finally, solve for x: x = ln(c/a) / ln(b). This method is fundamental for finding time, number of periods, or other exponential variables in practical problems.

Real-World Applications

Exponential functions are used to model various real-world phenomena. Common examples include population growth, where the population increases by a fixed percentage over time; compound interest, where the interest earned is added to the principal, leading to exponential growth of money; and radioactive decay, where the amount of a radioactive substance decreases by a fixed percentage over time. Understanding how to set up the exponential equation for these scenarios, identify 'a' and 'b', and solve for unknown variables is a key skill.