Graphs – Seeing Algebra in Pictures
Graphs give algebra a visual form. By plotting equations on a coordinate grid, we can see patterns, find solutions, and understand how two quantities relate to each other without solving equations by hand.
The Coordinate Grid
The coordinate grid has two axes: the horizontal x-axis and the vertical y-axis. Every point is described by an ordered pair (x, y). The point where the axes cross is the origin (0, 0).
Plotting Points
| Point | x | y | Quadrant |
|---|---|---|---|
| (3, 4) | 3 | 4 | Top right (1st) |
| (-2, 5) | -2 | 5 | Top left (2nd) |
| (-1, -3) | -1 | -3 | Bottom left (3rd) |
| (4, -2) | 4 | -2 | Bottom right (4th) |
Graphing a Linear Equation (Straight Line)
A linear equation produces a straight line. The form y = mx + c is the most useful, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).
Make a table: x=-1 gives y=-1; x=0 gives y=1; x=2 gives y=5. Plot these three points and draw a straight line through them. The gradient is 2 (rise 2 for every 1 right) and the y-intercept is (0, 1).
Gradient (Slope)
| Gradient Value | What the Line Does |
|---|---|
| Positive (e.g. 3) | Slopes upward left to right |
| Negative (e.g. -2) | Slopes downward left to right |
| Zero | Horizontal flat line |
| Very large (e.g. 10) | Nearly vertical, steep |
Graphing a Quadratic (Parabola)
A quadratic equation y = ax squared + bx + c produces a U-shaped (or upside-down U) curve called a parabola.
Make a table: x=-2 gives y=0; x=-1 gives y=-3; x=0 gives y=-4; x=1 gives y=-3; x=2 gives y=0. The parabola opens upward, with its lowest point (vertex) at (0, -4) and crossing the x-axis at x=-2 and x=2.
Finding Solutions from a Graph
The x-coordinates where a graph crosses the x-axis are the solutions of the equation when y = 0. These are called roots or x-intercepts.
Real-Life Application
- The distance a car travels over time is a straight-line graph (at constant speed).
- The path of a football is a parabola.
- A company profit graph helps managers see at which sales level profit is maximised.
Key Takeaways
- Every equation can be drawn as a graph, giving a visual picture of its solutions.
- Linear equations (y = mx + c) produce straight lines; m is the gradient and c is the y-intercept.
- Quadratic equations produce parabolas that open upward (a bigger than 0) or downward (a less than 0).
- Where the graph crosses the x-axis gives the solutions (roots) of the equation.
Practice Questions
- Plot the graph of y = 3x - 2 for x from -2 to 3.
- What is the gradient and y-intercept of y = -x + 4?
- A line passes through (0, 3) and (2, 7). Find its gradient and equation.
- Complete the table for y = x squared - 2x - 3 for x from -1 to 4, then plot the parabola and state the x-intercepts.
- Two lines are y = 2x - 1 and y = x + 2. Sketch both and find where they intersect.