# Graphs, Functions & Transformations

When sketching graphs, it is important to clearly show and label any coordinate-axis intercepts (y-intercepts and roots) as well as any stationary points (e.g. turning points).

### Linear Graphs

The general from for a linear graph is ** y = mx + c**, where m is the gradient and c the y-intercept. Gradient is found as rise/run:

This equation can be rearranged to give an alternate equation for a line, which is more useful when you know two points and need to know the line connecting them.

y2 - y1 = m(x2 - x1)

To find the length of a section of line, use **Pythagoras' Theorem.**

Two

**parallel lines**have an equal gradient, so will never meet.Two

**perpendicular lines**have gradients that are each other's negative reciprocal, and so they do cross. This means that**the product of their two gradients equals -1**

### Quadratic Graphs

The general form of a quadratic expression is **ax² + bx + c.**** **All quadratic graphs are parabola-shaped, symmetrical about one turning point (this can be a maximum or minimum):

For quadratics in the form

*ax² + bx + c*,**c****is the y-intercept.**Completing the square gives the coordinates of the turning point:

**When***f(x) = a(x + p)² + q*, the turning point is at (-p, q)The discriminant tells you how many roots there are, so how many times the graph crosses the x-axis.

### Cubic Graphs

The general form for a cubic expression is * ax³ + bx² + cx + d*, and can intercept the x-axis at 1,2 or 3 points.

If you do not know the coefficient, then you can find out which way the graph goes by seeing what happens **as x tends to ±∞:**

If as

**x**→**∞, y**→**∞**and**x**→ -**∞, y**→ -**∞**, the graph is**positive**If as

**x**→**∞, y**→ -**∞**and**x**→ -**∞, y**→**∞**, the graph is**negative**

Cubic graphs can have just 1 or 3 distinct roots, 1 distinct root with a repeated root, or 1 triple repeated root.

A

**triple repeated root**occurs when the graph has just one stationary point, and this is on the x-axisA

**distinct root**occurs when the graph crosses the x-axisA

**repeated root**occurs when the graph touches the x-axis but does not cross it

To sketch a cubic, you need to know the roots. If it is given in the form *ax³ + bx² + cx + d*, you need to factorise it first. This will tell you how many roots it has, and where they are. Then, testing to see what happens as x tends to ±∞ shows the shape.

### Quartic Graphs

The standard form for a quartic function is * ax⁴ + bx³ + cx² + dx + e* where a, b, c, d and e are real numbers and a is not zero.

Again, you need to know the roots of the function and the y-intercept to be able to sketch it.

### Reciprocal Graphs

To sketch graphs of reciprocals, such as **y = 1/x**, ** y = 1/x²**, or -

**, you need to know the asymptotes. These are lines that the graphs tend towards, but never touch or cross.**

*3/x*Graphs in the formy = k/x or y = k/x²have asymptotes at x=0 and y=0

The greater the value of the numerator, the further the graph is from the coordinate axis. The asymptotes are still y=0 and x=0, however.

## Functions

In maths, functions are relationships that map a value from a set of inputs to a single output. The set of inputs is known as the **domain**, and the set of possible outputs is the **range**. The **roots** of a function are the values of *x* for which *f(x) = 0*.

There are two types of functions, **one-to-one** and **many-to-one**. Anything else is **not a function**: