Graphs of functions

1. Overview

Graphs of functions are visual representations of mathematical relationships, showing how the output ($y$-value) changes with the input ($x$-value). Understanding graphs allows you to solve equations, identify key features like maximum and minimum points, and model real-world situations. This topic covers plotting graphs from equations, recognizing common graph shapes, and using graphs to solve equations.

Key Definitions

• Function: A mathematical relationship where each input ($x$) has a unique output ($y$).
• Gradient: The steepness of a line, calculated as $\frac{\text{Change in } y}{\text{Change in } x}$.
• Intercept: The point where a graph crosses an axis (y-intercept is where $x=0$; x-intercept/root is where $y=0$).
• Root: The $x$-value(s) where the graph crosses the x-axis (i.e., where $f(x) = 0$).
• Asymptote: A line that a graph approaches but never actually touches or crosses (common in reciprocal and exponential graphs).
• Stationary Point: A point on a curve where the gradient is zero (a maximum or minimum).

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Core Content

3.1 Constructing Tables and Drawing Graphs

To draw any graph, you must first create a table of values.

1. Choose or use the given range of $x$-values.
2. Substitute each $x$ into the function to find $y$.
3. Plot the $(x, y)$ coordinates on a Cartesian plane.
4. Join the points: use a ruler for linear graphs ($ax + b$) and a smooth curve for all others.

3.2 Common Graph Shapes (Core)

• Linear ($y = ax + b$): A straight line. $a$ is the gradient, $b$ is the y-intercept.
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• Quadratic ($y = x^2 + ax + b$): A U-shaped curve called a parabola.
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• Reciprocal ($y = \frac{k}{x}$): Two separate curves in opposite quadrants. It never touches $x=0$ or $y=0$.
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3.3 Solving Equations Graphically

To solve $f(x) = k$:

1. Draw the graph of $y = f(x)$.
2. Draw the horizontal line $y = k$.
3. The solutions (roots) are the $x$-coordinates where the two lines intersect.

Worked example 1 — Solving a quadratic graphically

Question: Draw the graph of $y = x^2 - 2x - 3$ for $-2 \leq x \leq 4$ and use it to solve $x^2 - 2x - 3 = 0$.

Step 1: Table of values

$x$	-2	-1	0	1	2	3	4
$y$	5	0	-3	-4	-3	0	5

Calculation for $x = -2$: $y = (-2)^2 - 2(-2) - 3$ $y = 4 + 4 - 3 = 5$

Step 2: Plot and Solve Plot the points and join with a smooth curve. To solve $x^2 - 2x - 3 = 0$, look at where the graph crosses the x-axis ($y=0$). Solutions: $x = -1$ and $x = 3$.

Worked example 2 — Solving a linear equation graphically

Question: Draw the graph of $y = 2x - 1$ for $-1 \leq x \leq 3$. Use your graph to solve $2x - 1 = 3$.

Step 1: Table of values

$x$	-1	0	1	2	3
$y$	-3	-1	1	3	5

Calculation for $x = -1$: $y = 2(-1) - 1$ $y = -2 - 1 = -3$

Step 2: Plot the graph Plot the points and join them with a straight line using a ruler.

Step 3: Solve $2x - 1 = 3$ Draw the horizontal line $y = 3$ on the same graph. Find the x-coordinate of the point where the line $y = 3$ intersects the line $y = 2x - 1$.

Solution: $x = 2$

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Extended Content (Extended Only)

4.1 Advanced Power Functions ($ax^n$)

You must recognize shapes for various values of $n$:

• $n = 3$ (Cubic): $y = x^3$. An S-shaped curve passing through the origin.
• $n = \frac{1}{2}$ (Square Root): $y = \sqrt{x}$. Only exists for $x \geq 0$.
• $n = -1$ (Reciprocal): $y = \frac{1}{x}$. (See Core).
• $n = -2$ (Inverse Square): $y = \frac{1}{x^2}$. Both parts of the curve are above the x-axis.
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Worked example 3 — Sketching an inverse square function

Question: Sketch the graph of $y = \frac{2}{x^2}$.

Step 1: Consider the behavior as x approaches 0 As $x$ gets closer to 0, $y$ becomes very large. The y-axis is a vertical asymptote.

Step 2: Consider the behavior as x approaches infinity As $x$ gets very large (positive or negative), $y$ approaches 0. The x-axis is a horizontal asymptote.

Step 3: Note the symmetry Since $x$ is squared, the function is symmetrical about the y-axis.

Step 4: Sketch the graph Draw a curve in the first quadrant that approaches both axes but never touches them. Draw a symmetrical curve in the second quadrant. Both curves should be above the x-axis.

4.2 Exponential Graphs ($ab^x + c$)

Used for growth ($b > 1$) or decay ($0 < b < 1$).

• The graph $y = 2^x$ increases rapidly as $x$ increases.
• The line $y = c$ is the horizontal asymptote.

Worked example 4 — Finding the intersection of an exponential and a linear function

Question: Find the intersection of $y = 2^x$ and $y = 3 - x$.

Step 1: Table of values for $y = 2^x$

$x$	-1	0	1	2
$y$	0.5	1	2	4

Step 2: Table of values for $y = 3 - x$

$x$	-1	0	1	2
$y$	4	3	2	1

Step 3: Plot both functions Draw the curve for $y = 2^x$ and the straight line for $y = 3 - x$. The intersection point is approximately $x = 1$, $y = 2$. Solution: $x \approx 1$.

4.3 Determining the Nature of Stationary Points

If asked to prove if a point is a maximum or minimum:

1. Find the second derivative $\frac{d^2y}{dx^2}$.
2. Substitute the $x$-value of the stationary point.
3. If $\frac{d^2y}{dx^2} > 0$, it is a Minimum.
4. If $\frac{d^2y}{dx^2} < 0$, it is a Maximum.

Worked example 5 — Determining the nature of a stationary point

Question: The function $y = x^3 - 3x^2 + 2$ has a stationary point at $x = 0$. Determine whether this point is a maximum or a minimum.

Step 1: Find the first derivative $\frac{dy}{dx} = 3x^2 - 6x$

Step 2: Find the second derivative $\frac{d^2y}{dx^2} = 6x - 6$

Step 3: Substitute the x-value of the stationary point Substitute $x = 0$ into the second derivative: $\frac{d^2y}{dx^2} = 6(0) - 6 = -6$

Step 4: Determine the nature of the stationary point Since $\frac{d^2y}{dx^2} = -6 < 0$, the stationary point at $x = 0$ is a Maximum.

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Key Equations

Linear Gradient: $m = \frac{y_2 - y_1}{x_2 - x_1}$ (Not on formula sheet)

Linear Equation: $y = mx + c$ (Not on formula sheet)

Quadratic Form: $y = ax^2 + bx + c$ (Not on formula sheet)

Exponential Growth/Decay: $y = Ab^x$ (Not on formula sheet)

Note: These formulas are NOT provided on the IGCSE formula sheet; they must be memorized.

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Common Mistakes to Avoid

• ❌ Incorrect Gradient Calculation: Calculating the gradient by simply counting squares on the graph without considering the scale of the axes.
  • ✓ Correct Gradient Calculation: Use the coordinates of two points on the line and the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, paying close attention to the scale on each axis. For example, if each square on the x-axis represents 0.5 units and each square on the y-axis represents 2 units, account for this in your calculation.
• ❌ Joining Non-Linear Points with a Ruler: Drawing a quadratic or cubic graph by connecting the plotted points with straight lines.
  • ✓ Drawing Smooth Curves: Join the points with a smooth, continuous curve. The curve should pass through all the points and not have any sharp corners.
• ❌ Ignoring the Negative Sign in Substitution: Incorrectly substituting negative values into a function, especially when squaring. For example, calculating $(-2)^2$ as $-4$ instead of $4$.
  • ✓ Using Brackets for Negative Numbers: Always use brackets when substituting negative numbers into a function, especially when using a calculator. For example, write $(-2)^2$ to ensure the calculator squares the entire negative number.
• ❌ Incorrectly Identifying Roots: Reading the roots of the equation $f(x) = k$ as the y-coordinates of the intersection points instead of the x-coordinates.
  • ✓ Correctly Identifying Roots: Remember that the roots are the x-values where the graph of $y = f(x)$ intersects the line $y = k$.

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Exam Tips

• Command Words:
  • "Plot": Precisely place points and join them.
  • "Sketch": Draw the general shape and label intercepts (no grid needed).
• Calculator Tip: Use the "Table" mode on your scientific calculator to generate $y$-values quickly for a given function. This prevents manual calculation errors.
• Scale: Check the scale of the axes carefully. Sometimes 1 unit is 2cm on the x-axis but 1cm on the y-axis.
• Real-world Context: Expect exponential graphs in questions about bacteria growth, compound interest, or radioactive decay. The y-intercept usually represents the "initial amount."
• Accuracy: When solving graphically, your answer should be within $\pm 0.1$ of the exact value. Always draw the construction lines (the dashed lines from the graph to the axes) to show the examiner where your answer came from.