Graphs of functions - Mathematics (0580) - Cambridge IGCSE

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Welcome to Graphs of Functions!

Graphs are one of the most powerful tools in Mathematics. They allow us to take abstract equations (like $y = 2x + 3$) and see them as shapes and lines, helping us understand relationships instantly!

In this chapter, we will learn how to recognise different types of function graphs, sketch them efficiently, and use your graphic display calculator (GDC) like a pro to find key features. Don't worry if sketching curves seems tricky; we break down every shape into simple steps.

Section 1: The Foundations of Functions and Graphs

1.1 Understanding Function Notation and Coordinates

When we talk about graphs in IGCSE, we usually talk about plotting a function.

A function is a rule that assigns exactly one output value (y) for every input value (x).
We often write a function as $f(x)$ (read as "f of x"). Remember, $f(x)$ is just another way of writing $y$.
Example: If $f(x) = 3x - 1$, then $y = 3x - 1$.

Plotting Points:

Every point on a graph is described by Cartesian coordinates $(x, y)$.

If you are given a function, you can always generate a table of values to plot the graph manually.

Step-by-step example: Plotting $f(x) = x^2$ for $x$ values $-2$ to $2$.

Choose your x-values: $-2, -1, 0, 1, 2$.
Calculate the corresponding $y$ or $f(x)$ value:
- $f(-2) = (-2)^2 = 4 \rightarrow (-2, 4)$
- $f(0) = (0)^2 = 0 \rightarrow (0, 0)$
- $f(2) = (2)^2 = 4 \rightarrow (2, 4)$
Plot these points $(x, y)$ onto your coordinate grid.
Join the points with a smooth curve (for non-linear functions) or a straight line (for linear functions).

Quick Review: Core vs. Extended Content

The functions topic is tiered. All students must recognize linear and quadratic graphs. Extended students need to recognize cubic, reciprocal, exponential, and trigonometric graphs as well.

Section 2: Recognizing Shapes of Graphs (C3.1, E3.1)

The first step in graphing is recognising the function type from its equation and knowing the general shape it produces.

2.1 Linear Functions

Form: $f(x) = ax + b$ (or $y = mx + c$)

Shape: A straight line.

The value $a$ (or $m$) is the gradient (steepness).
The value $b$ (or $c$) is the y-intercept (where the line crosses the y-axis).
If $a$ is positive, the line slopes up (positive gradient).
If $a$ is negative, the line slopes down (negative gradient).

Analogy: Think of climbing a hill. If the gradient is steep ($a$ is large), the climb is hard!

Key Takeaway: If the highest power of $x$ is $x^1$, it's a straight line.

2.2 Quadratic Functions (The Parabola)

Form: $f(x) = ax^2 + bx + c$

Shape: A parabola (a U-shape or an inverted n-shape).

If $a > 0$ (positive), the parabola is U-shaped (happy face, minimum point).
If $a < 0$ (negative), the parabola is n-shaped (sad face, maximum point).
The highest or lowest point is called the vertex or turning point.

Extended Focus: Finding the Quadratic Equation (E3.4)

If you are given the graph, you might need to find its equation. The most useful form is the vertex form:

$$y = a(x - h)^2 + k$$

Here, the vertex is $(h, k)$.

Example: A parabola has a vertex at $(3, -1)$ and passes through the point $(4, 1)$.

Substitute the vertex $(h, k) = (3, -1)$:
$y = a(x - 3)^2 - 1$
Substitute the given point $(4, 1)$ to find $a$:
$1 = a(4 - 3)^2 - 1$
$1 = a(1)^2 - 1$
$2 = a$
The equation is: $y = 2(x - 3)^2 - 1$

Key Takeaway: The vertex form tells you the turning point directly, which is crucial for sketching.

2.3 Extended Graph Types (E3.1)

Cubic Functions

Form: $f(x) = ax^3 + bx^2 + cx + d$

Shape: Generally, an S-shape or an elongated Z-shape. It can have up to two turning points (a local maximum and a local minimum).

Reciprocal Functions

Form: $f(x) = \frac{k}{x}$

Shape: A hyperbola consisting of two separate branches (in opposite quadrants). These graphs never touch the axes.

Exponential Functions

Form: $f(x) = a^x$ (where $a > 0$)

Shape: A curve that grows or decays incredibly fast.

If $a > 1$: Exponential Growth (e.g., population increase). The curve rapidly increases as $x$ increases.
If $0 < a < 1$: Exponential Decay (e.g., radioactive decay). The curve rapidly decreases as $x$ increases.

Did you know? Exponential functions are used to model compound interest—money that grows based on both the original amount AND the interest it has already earned!

Trigonometric Functions

Forms: $f(x) = a \sin(bx)$, $f(x) = a \cos(bx)$, and $\tan x$.

Shape: Periodic (repeating patterns).

Sine ($\sin x$) and Cosine ($\cos x$) graphs look like smooth waves.
The amplitude (maximum height from the middle line) is given by $a$.
The period (how long it takes for the wave to repeat) is influenced by $b$.
Tangent ($\tan x$) looks very different, consisting of many separate sections, with lines it never touches (asymptotes) at $90^\circ, 270^\circ$, etc.

Section 3: Key Features of Graphs

3.1 Intercepts and Zeros (C3.2, E3.2)

Knowing where a graph crosses the axes is essential for sketching.

Y-intercept: This is where the graph crosses the y-axis. This happens when $x = 0$.
X-intercepts (or Zeros/Roots): This is where the graph crosses the x-axis. This happens when $y = 0$ (i.e., when $f(x) = 0$).

Tip: Finding zeros usually means solving an equation. For quadratics, you can factorise or use the quadratic formula. For complex functions, you must use your GDC.

3.2 Turning Points: Maxima and Minima (C3.2, E3.2)

Turning points are where the graph changes direction (e.g., from going down to going up).

A local maximum is a peak (the highest point in a small area).
A local minimum is a trough (the lowest point in a small area).

For IGCSE, especially for unfamiliar functions, you will use your Graphic Display Calculator (GDC) to find the exact coordinates of these points (C3.2/E3.2).

3.3 Asymptotes (Extended Only: E3.5)

An asymptote is a line that a graph approaches but never actually touches, no matter how far out you trace the curve.

Analogy: Imagine a piece of sticky tape stuck to a table. The line is the tape, and the graph is a piece of paper getting closer and closer to the tape without ever lying perfectly flat on it.

Vertical Asymptotes: Occur when the function is undefined, often due to division by zero.
Example: For $f(x) = \frac{1}{x}$, the vertical asymptote is $x=0$.
Horizontal Asymptotes: Describe the long-term behaviour of the graph as $x$ becomes very large (positive or negative).
Example: For $f(x) = \frac{1}{x}$, the horizontal asymptote is $y=0$.
Trigonometric Example: The graph of $y = \tan x$ has vertical asymptotes at $x = 90^\circ$, $x = 270^\circ$, etc.

Key Takeaway: When sketching graphs that have asymptotes (like reciprocal or tan graphs), make sure your sketch approaches the line but never crosses it.

Section 4: Using the Graphic Display Calculator (GDC) (C3.2, E3.2)

The GDC is an essential tool for graphing and solving complex function problems quickly.

4.1 Core GDC Functions

You must be proficient in using your GDC for the following tasks (for any function, even unfamiliar ones):

Sketch the graph: Enter the function into the Y= editor and view the plot. Ensure your viewing window (Zoom settings) captures the important features.
Produce a table of values: Generate specific coordinate pairs to help you plot points accurately on paper.
Plot points: Use the table data to mark points (usually with a small cross, $\times$) onto the graph paper.
Find zeros (x-intercepts): Use the "Calculate" menu (or equivalent function) to find where $y=0$.
Find local maxima or local minima: Use the "Calculate" menu to find the coordinates of turning points.
Find the intersection point of two graphs: Enter both functions (Y1 and Y2) and use the "Calculate Intersection" tool.
Find the vertex of a quadratic: This is just a special case of finding the maximum or minimum point (Step 5).

4.2 Solving Equations Graphically (E2.5, E3.2)

Your GDC is vital for solving equations that are hard to solve algebraically, like $2x = x^2$ or $2x - 1 = \frac{1}{x}$.

Step-by-step method for solving equations using intersection:

Rewrite the equation so you have two separate functions, $Y_1$ and $Y_2$.
Example: To solve $2x - 1 = \frac{1}{x}$, set $Y_1 = 2x - 1$ and $Y_2 = \frac{1}{x}$.
Enter $Y_1$ and $Y_2$ into your GDC and sketch the graphs.
Use the "Calculate Intersection" feature on your calculator.
The $x$-coordinates of the intersection points are the solutions to the original equation.

Key Takeaway: Practice these GDC skills repeatedly. In the exam, time saved by efficient calculator use is crucial!

Section 5: Transforming Graphs (Extended Only: E3.6)

Graph transformations involve taking a basic function, $y = f(x)$, and moving it around the axes. The syllabus focuses specifically on translations (shifts).

5.1 Vertical Translation: Up and Down

When the transformation is in the form:

$$y = f(x) + k$$

If $k$ is positive, the graph shifts $k$ units up.
If $k$ is negative, the graph shifts $k$ units down.

This translation affects the $y$-coordinate of every point, including the y-intercepts and turning points.

Example: If $f(x) = x^2$, then $y = f(x) + 3 = x^2 + 3$ shifts the graph 3 units up.

5.2 Horizontal Translation: Left and Right

When the transformation is in the form:

$$y = f(x + k)$$

This is often counter-intuitive because the movement is the opposite direction to the sign of $k$.

If $k$ is positive (e.g., $f(x+3)$), the graph shifts $k$ units left (negative x-direction).
If $k$ is negative (e.g., $f(x-3)$), the graph shifts $k$ units right (positive x-direction).

Memory Aid: "LION" (Left Is ON the Opposite side). If the number is inside the brackets with the $x$, the shift is horizontal and the direction is the opposite of the sign.

Example: The graph of $y = (x-2)^2$ is the graph of $y = x^2$ shifted 2 units to the right.

Key Takeaway: A transformation outside the brackets $(+k)$ moves the graph vertically (as expected). A transformation inside the brackets $(x+k)$ moves the graph horizontally (opposite direction).

Section 6: Exponential and Logarithmic Functions (Extended Only: E3.7)

This section connects two important functions: the exponential function and its inverse, the logarithmic function.

6.1 The Logarithmic Function

The logarithmic function is essentially the reverse process of the exponential function.

The relationship between the exponential form and the logarithmic form is:

$$y = a^x \quad \text{is equivalent to} \quad x = \log_a y$$

We read $x = \log_a y$ as "x is the logarithm of y to the base a."

For IGCSE, unless another base is specified, all logarithms are base 10. This means $\log y$ is usually $\log_{10} y$.

Solving Equations with Logs

A key use of logarithms is solving for an unknown power (index).

If you have an equation like $a^x = b$, the solution for $x$ is given by:

$$x = \frac{\log b}{\log a}$$

Example: Solve $5^x = 100$.

$$x = \frac{\log 100}{\log 5}$$ $$x = \frac{2}{\log 5}$$ (Use your calculator to find the numerical value.)

6.2 Inverse Functions (E3.3)

The concept of a logarithm being the inverse of an exponential function links to inverse functions generally.

If $y = f(x)$, the inverse function, written as $f^{-1}(x)$, reverses the process.

Step-by-step: Finding the inverse function

Write the function as $y = f(x)$.
Swap $x$ and $y$.
Rearrange the new equation to make $y$ the subject.
Replace $y$ with $f^{-1}(x)$.

Example: Find the inverse of $f(x) = 2x + 5$.
1. $y = 2x + 5$
2. $x = 2y + 5$ (Swapping)
3. $x - 5 = 2y \rightarrow y = \frac{x - 5}{2}$ (Rearranging)
4. $f^{-1}(x) = \frac{x - 5}{2}$

Graph Connection: The graph of $y = f^{-1}(x)$ is a reflection of the graph of $y = f(x)$ in the line $y = x$.

Key Takeaway: Logarithms and Graphs

Logarithmic functions and exponential functions are inverses. You may be asked to solve exponential or logarithmic equations by reading values directly from a plotted graph (E3.7).