Sketching Linear Graphs: Your Ultimate Revision Guide (Corbettmaths Included)

Welcome to eokultv's Ultimate Revision Guide: Sketching Linear Graphs!

As an expert educator, I'm thrilled to help you master the art of sketching linear graphs. This guide is designed to be comprehensive, ensuring you're fully prepared, with insights that complement resources like Corbettmaths.

Definition: What is a Linear Graph?

A linear graph is the visual representation of a linear relationship between two variables, typically $x$ and $y$. When plotted on a Cartesian coordinate plane, this relationship always forms a straight line. The general equation of a straight line is often expressed in the gradient-intercept form:

$$y = mx + c$$

• $y$ and $x$ represent the variables plotted on the vertical and horizontal axes, respectively.
• $m$ represents the gradient (or slope) of the line. It tells us how steep the line is and its direction. A positive $m$ means the line slopes upwards from left to right, while a negative $m$ means it slopes downwards. The gradient is calculated as the 'rise' over the 'run' (change in $y$ over change in $x$).
• $c$ represents the y-intercept. This is the point where the line crosses the y-axis, meaning the value of $y$ when $x = 0$.

History and Background: The Dawn of Coordinate Geometry

The ability to represent algebraic equations geometrically on a graph is largely attributed to the French mathematician and philosopher René Descartes in the 17th century. His groundbreaking work, particularly 'La Géométrie', led to the development of what we now call the Cartesian coordinate system (named after him). This innovation merged geometry and algebra, allowing for equations to be visualized and geometric shapes to be described algebraically. This revolutionary concept paved the way for calculus and has become fundamental across all scientific and engineering disciplines.

Key Principles: How to Sketch Linear Graphs Effectively

Sketching a linear graph requires identifying at least two points that lie on the line, or one point and the line's gradient. Here are the primary methods, often emphasized in resources like Corbettmaths:

Method 1: Using the Gradient-Intercept Form ($y = mx + c$)

This is arguably the most common and intuitive method, heavily featured in Corbettmaths tutorials. If your equation isn't in this form, rearrange it first.

1. Identify the y-intercept ($c$): This is your first point on the graph, $(0, c)$. Plot this point on the y-axis.
2. Identify the gradient ($m$): Remember $m = \frac{\text{rise}}{\text{run}}$.
   • If $m$ is a whole number (e.g., $m=2$), think of it as $\frac{2}{1}$. From your y-intercept, move 1 unit to the right (run) and 2 units up (rise) to find your second point.
   • If $m$ is a fraction (e.g., $m = \frac{2}{3}$), from your y-intercept, move 3 units to the right (run) and 2 units up (rise).
   • If $m$ is negative (e.g., $m = -2$ or $m = -\frac{1}{2}$), the 'rise' becomes a 'fall'. So for $m = -2 = \frac{-2}{1}$, move 1 unit right and 2 units down.
3. Draw the line: Once you have two distinct points, use a ruler to draw a straight line through them, extending it across your coordinate plane. Don't forget to label the line with its equation.

Method 2: Using Two Points (Intercept Method)

This method is particularly useful when the equation is not easily rearranged into $y=mx+c$ or when finding the intercepts is straightforward. Corbettmaths also frequently demonstrates this approach.

1. Find the y-intercept: Set $x = 0$ in the equation and solve for $y$. This gives you the point $(0, y_{\text{intercept}})$. Plot this point.
2. Find the x-intercept: Set $y = 0$ in the equation and solve for $x$. This gives you the point $(x_{\text{intercept}}, 0)$. Plot this point on the x-axis.
3. Draw the line: Connect these two intercept points with a straight line, extending it and labelling it.

Special Cases of Linear Graphs

• Horizontal Lines ($y = k$): These lines have a gradient of $m=0$. They are parallel to the x-axis and pass through the point $(0, k)$ on the y-axis. Example: $y=3$.
• Vertical Lines ($x = k$): These lines have an undefined gradient. They are parallel to the y-axis and pass through the point $(k, 0)$ on the x-axis. Example: $x=-2$.
• Lines through the Origin ($y = mx$): These lines have a y-intercept of $c=0$, meaning they pass through the origin $(0,0)$. You then use the gradient $m$ to find a second point.

Summary of Steps for Sketching (Corbettmaths Style)

For most linear graphs, especially those for examinations:

1. Rearrange the equation into the $y = mx + c$ form if it isn't already.
2. Identify the y-intercept ($c$) and plot $(0, c)$.
3. Identify the gradient ($m$). Use 'rise over run' to find a second point.
4. (Alternative/Check) Find the x-intercept by setting $y=0$ and solving for $x$.
5. Draw a clear, straight line extending through your chosen points, using a ruler.
6. Label the line with its equation and ensure axes are labelled if not already.

Real-world Examples of Linear Graphs

Linear relationships are ubiquitous in the real world, providing simple yet powerful models for various phenomena:

Conclusion: Mastering the Straight Line

Sketching linear graphs is a foundational skill in mathematics, bridging algebra and geometry. By understanding the roles of the gradient ($m$) and the y-intercept ($c$), and practicing both the gradient-intercept method and the two-point intercept method, you'll be able to confidently visualize and interpret linear relationships. Remember that resources like Corbettmaths offer excellent practice questions and video explanations to solidify your understanding. Consistent practice is key to turning these principles into second nature!

Keep sketching, keep learning, and you'll master linear graphs in no time!