Functions, transformations and circle graphs (Higher)

Introduction to Functions and Graphs

Welcome to one of the most exciting parts of Higher Tier Algebra! In previous chapters, you learned how to solve equations and draw basic graphs. In this chapter, we take those skills further. We will look at functions as "mathematical machines," learn how to transform (move and flip) graphs, and master the geometry of circle graphs. These concepts are vital for understanding how different parts of mathematics connect.

Quick Review: Before starting, remember that in algebra, we use letters to represent numbers. In this chapter, we use special notation like \(f(x)\) to show that one value depends on another.

1. Understanding Functions

Think of a function as a machine. You put a number in (the input), the machine follows a rule, and a new number comes out (the output).

We use the notation \(f(x)\) to describe this. For example, if the rule is "double the number and add 3," we write:
\(f(x) = 2x + 3\)

Composite Functions

A composite function is what happens when you use two machines in a row. The output of the first machine becomes the input for the second. This is written as \(fg(x)\) or \(gf(x)\).

Important: Always work from the inside out. For \(fg(x)\), you apply function \(g\) first, then put that answer into function \(f\).

Example: If \(f(x) = x^2\) and \(g(x) = x + 1\):
To find \(fg(3)\):
1. Find \(g(3)\): \(3 + 1 = 4\)
2. Put that into \(f\): \(f(4) = 4^2 = 16\)
So, \(fg(3) = 16\).

Inverse Functions

An inverse function, written as \(f^{-1}(x)\), is the "undo" button. It takes the output and tells you what the original input was.

How to find the inverse:
1. Write the function as \(y = ...\)
2. Swap the \(x\) and \(y\).
3. Rearrange the equation to make \(y\) the subject.
4. Replace \(y\) with \(f^{-1}(x)\).

Key Takeaway: Functions are just rules. Composite functions are rules applied in sequence (inside to outside), and inverse functions reverse the rule.

2. Transforming Functions

You can change the position or orientation of any graph \(y = f(x)\) by slightly changing its equation. At the Higher level, you need to know translations and reflections.

Translations (Shifting the graph)

• Vertical Shift: \(y = f(x) + a\). This moves the graph up by \(a\) units. If \(a\) is negative, it moves down.
• Horizontal Shift: \(y = f(x + a)\). This moves the graph left or right.
  Warning: This is counter-intuitive! \(y = f(x + 2)\) moves the graph 2 units to the left, and \(y = f(x - 2)\) moves it 2 units to the right.

Reflections (Flipping the graph)

• Reflection in the x-axis: \(y = -f(x)\). All the \(y\)-values swap signs, flipping the graph upside down.
• Reflection in the y-axis: \(y = f(-x)\). All the \(x\)-values swap signs, flipping the graph horizontally.

Memory Trick: If the change is outside the bracket (like \(+ a\) or the minus sign in \(-f(x)\)), it affects the vertical direction (\(y\)). If the change is inside the bracket, it affects the horizontal direction (\(x\)) and often does the "opposite" of what you expect!

3. Circle Graphs

In this section of the Algebra curriculum, we look at circles centered at the origin \((0,0)\).

The general equation for a circle with radius \(r\) is:
\(x^2 + y^2 = r^2\)

Example: If you see the equation \(x^2 + y^2 = 25\), the radius is not 25. The radius is \(\sqrt{25} = 5\).

Finding the Equation of a Tangent

A tangent is a straight line that touches the circle at exactly one point. A key rule from circle theorems is that the tangent is perpendicular to the radius at that point.

Step-by-step: How to find the equation of a tangent at point \((x_1, y_1)\):
1. Find the gradient of the radius (the line from \((0,0)\) to your point) using \(m = \frac{y_1 - 0}{x_1 - 0}\).
2. Find the gradient of the tangent. Since they are perpendicular, the tangent's gradient is the "negative reciprocal" (\(\frac{-1}{m}\)).
3. Use the straight-line formula \(y - y_1 = m(x - x_1)\) with your tangent gradient and the point coordinates to find the final equation.

Quick Review: For a circle \(x^2 + y^2 = r^2\), the center is always \((0,0)\). To find the tangent, remember: Radius gradient \(\times\) Tangent gradient \(= -1\).

4. Common Pitfalls and Tips

• Don't forget the square root: In circle equations, the number on the right is \(r^2\). Always square root it to find the actual radius.
• Order matters in composites: \(fg(x)\) is usually very different from \(gf(x)\). Always start with the function closest to the \(x\).
• The "Inside-Opposite" Rule: For transformations, changes inside the bracket \(f(x+a)\) affect the \(x\)-axis and work in the opposite direction to the sign.
• Perpendicular Gradients: If the radius gradient is \(\frac{2}{3}\), the tangent gradient must be \(-\frac{3}{2}\).

Did you know? These transformations don't just work for lines; they work for every type of graph you've learned—quadratics, cubics, and even trigonometric graphs like sine and cosine!

Summary of Key Formulae

Circle Equation: \(x^2 + y^2 = r^2\)
Perpendicular Gradients: \(m_1 \times m_2 = -1\)
Translation (Up): \(f(x) + k\)
Translation (Left): \(f(x + k)\)
Reflection (x-axis): \(-f(x)\)
Reflection (y-axis): \(f(-x)\)

Don't worry if this seems tricky at first! Practicing the step-by-step methods for finding inverse functions and tangent equations will make these processes feel like second nature.