Graphs Revision Notes

The Coordinate Plane and Axes

The coordinate plane is formed by two perpendicular number lines: the x-axis and the y-axis. The x-axis is horizontal, and the y-axis is vertical. Their intersection point is called the origin, with coordinates (0, 0). It's crucial to remember that positive x-values are to the right of the origin, and negative x-values are to the left. Similarly, positive y-values are above the origin, and negative y-values are below.

Writing and Reading Coordinates (x, y)

Coordinates are always written as an ordered pair (x, y). The 'x' value represents the horizontal position relative to the origin, and the 'y' value represents the vertical postion. When reading or plotting, always go 'along the corridor (x-axis) and then up or down the stairs (y-axis)'. This mnemonic helps ensure you remember the correct order of x then y.

Plotting Points on a Grid

To plot a point (a, b) on a grid: Start at the origin (0, 0). Move 'a' units horizontally along the x-axis. If 'a' is positive, move right; if negative, move left. From that position, move 'b' units vertically along the y-axis. If 'b' is positive, move up; if negative, move down. Mark the final position clearly. Accuracy in plotting is essential for understanding geometric relationships.

What is a Graph of an Equation?

A graph of an equation is a visual representation of all the ordered pairs (x, y) that satisfy the equation. For example, if you have the equation y = x + 2, every point on its line, such as (1, 3) or (0, 2), will make the equation true when you substitute its x and y values. Understanding this allows you to see patterns and relationships between variables quickly.

Plotting Points from an Equation

To plot points for an equation, follow these steps: 1. Choose a suitable range of x-values (e.g., from -2 to 2). 2. For each chosen x-value, substitute it into the equation to calculate the corresponding y-value. 3. This gives you a coordinate pair (x, y). For instance, for y = 3x - 1, if x=1, y = 3(1)-1 = 2, so the point is (1, 2). 4. Repeat this for several points to get a good idea of the graph's shape.

Drawing Lines and Curves

After plotting enough points, the next step is to connect them. If the equation is linear (e.g., y = mx + c), the points will form a straight line. Use a ruler to draw a neat, continuous line through them, extending it across the axes if appropriate. For non-linear equations (e.g., y = x² or y = 1/x), the points will form a curve. Draw a smooth curve freehand through the plotted points. Ensure your line or curve extends sufficiently to show the behaviour of the function within the given range or context.

Interpreting Graphs

Graphs can be used to interpret information about equations. For example, the point where a graph crosses the x-axis is called an x-intercept (where y=0), and where it crosses the y-axis is called a y-intercept (where x=0). For linear graphs, the 'm' in y = mx + c represents the gradient (steepness), and 'c' is the y-intercept. Being able to read these features directly from a graph is a key skill.

Plotting Linear Graphs

To plot a linear graph for an equation like y = mx + c, you need to find at least two pairs of (x, y) coordinates that satisfy the equation. A good practice is to find three points to ensure accuracy and to catch any calculation errors. Choose simple x-values, such as -1, 0, 1, or 2, substitute them into the equation to find the corresponding y-values, then plot these points on a coordinate grid and connect them with a straight line. Remember to label your axes.

Interpreting y = mx + c

The standard form of a linear equation is y = mx + c. In this equation, 'm' represents the gradient of the line, which tells you how steep the line is and its direction (positive 'm' for upward slope from left to right, negative 'm' for downward slope). The 'c' represents the y-intercept, which is the point where the line crosses the y-axis. This means when x = 0, y = c. These two values provide key information about the relationship being graphed.

Finding Coordinates from an Equation

Given an equation, such as y = 3x - 2, you can find any number of (x, y) coordinate pairs by choosing a value for 'x' and then calculating the corresponding 'y' value. For example, if x = 0, y = 3(0) - 2 = -2, so the point is (0, -2). If x = 2, y = 3(2) - 2 = 6 - 2 = 4, so the point is (2, 4). These points lie on the line represented by the equation.

Recognising Linear vs. Non-linear Graphs

A linear graph will always be a straight line. This means the highest power of 'x' in the equation is 1 (e.g., y = 2x + 5). If the equation involves 'x' to a higher power (e.g., y = x² + 3), or operations like square roots or division by x, the graph will be non-linear, meaning it will be a curve. Always check the equation's structure to predict the graph's shape.

Understanding Asymptotes for Reciprocal and Exponential Functions

Asymptotes are lines that a graph approaches but never quite touches. For reciprocal functions like y = 1/x, there is a vertical asymptote at x = 0 (the y-axis) and a horizontal asymptote at y = 0 (the x-axis). This is because as x gets very close to 0, 1/x gets infinitely large or small, and as x gets very large (positive or negative), 1/x gets very close to 0. For exponential functions like y = a^x (where a > 0, a ≠ 1), there is always a horizontal asymptote at y = 0. This is because as x approaches negative infinity, a^x approaches 0. Recognising and sketching asymptotes is vital for accurately representing these functions.

Advanced Graph Transformations

Building on basic transformations, higher methods involve a deeper understanding of how algebraic changes affect a graph. For a function y = f(x):

1. y = f(x) + a: Translates the graph 'a' units vertically upwards (if a > 0) or downwards (if a < 0).
2. y = f(x + a): Translates the graph 'a' units horizontally to the left (if a > 0) or to the right (if a < 0).
3. y = a f(x): Stretches the graph vertically by a scale factor of 'a' (if a > 0). If 'a' is negative, it also reflects the graph in the x-axis.
4. y = f(ax): Stretches the graph horizontally by a scale factor of '1/a' (if a > 0). If 'a' is negative, it also reflects the graph in the y-axis.

Combining these transformations requires careful order of application, typically reflections/stretches before translations.

Solving Equations Graphically and Interpreting Intersections

The intersection points of two graphs y = f(x) and y = g(x) represent the solutions to the equation f(x) = g(x). For example, finding the roots of a quadratic equation ax^2 + bx + c = 0 is equivalent to finding where the graph y = ax^2 + bx + c intersects the x-axis (where y = 0). Similarly, setting f(x) = g(x) allows you to find where two functions' graphs meet. If an equation is rearranged into the form f(x) = k, the solutions correspond to the intersections of y = f(x) and the horizontal line y = k. This method is particularly useful for visualising the number of solutions or approximating their values directly from a graph.

Interpreting Gradients and Rates of Change

The gradient of a curve at a specific point on a graph represents the instantaneous rate of change of the y-value with respect to the x-value at that point. For linear functions, the gradient is constant. For non-linear functions, the gradient changes along the curve. A steeper gradient indicates a faster rate of change. For example, in a distance-time graph, the gradient represents speed; in a velocity-time graph, the gradient represents acceleration. Understanding how the gradient changes (e.g., increasing, decreasing, zero at turning points) provides crucial insights into the behaviour of the function and the real-world scenario it models. This concept forms the foundation for calculus.