Number & Algebra
Linear and Non-Linear Relationships
Year 9
- ✓By the end of this lesson students will be able to identify and graph linear relationships on the Cartesian plane.
- ✓By the end of this lesson students will be able to graphically solve simple simultaneous linear equations.
- ✓By the end of this lesson students will be able to recognise and sketch parabolas from quadratic equations.
- ✓By the end of this lesson students will be able to distinguish between linear and non-linear relationships based on their equations and graphs.
Key concepts
A linear relationship is a relationship between two variables that, when plotted on a Cartesian plane, forms a straight line. The rate of change (gradient) between the variables is constant. The general form of a linear equation is y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
To graph a linear equation, you can use a table of values by choosing several x-values and calculating their corresponding y-values, then plotting these points and drawing a straight line through them. Alternatively, you can use the gradient (m) and y-intercept (c): plot the y-intercept (0, c), then use the gradient (rise/run) to find a second point.
Simultaneous linear equations involve two or more linear equations with the same variables. The solution to a pair of simultaneous linear equations is the point (x, y) where their graphs intersect. This point satisfies both equations. Graphically, you plot both lines on the same Cartesian plane and identify their point of intersection.
A non-linear relationship is one where the graph is not a straight line. A common type of non-linear relationship encountered in Year 9 is a quadratic relationship, which graphs as a parabola. The general form of a quadratic equation is y = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' cannot be zero. Parabolas are symmetrical U-shaped or inverted U-shaped curves.
To graph a parabola, create a table of values by selecting a range of x-values (including positive, negative, and zero) and calculating their corresponding y-values. Plot these points and draw a smooth curve through them. Key features to note are the y-intercept (where x=0), x-intercepts (where y=0), and the vertex (the turning point of the parabola). If 'a' > 0, the parabola opens upwards; if 'a' < 0, it opens downwards.
Key facts to remember
- 1Linear graphs are always straight lines and have a constant gradient.
- 2The equation y = mx + c represents a linear relationship, where 'm' is the gradient and 'c' is the y-intercept.
- 3The solution to simultaneous linear equations is the point of intersection of their graphs.
- 4Parabolas are the graphs of quadratic equations (y = ax² + bx + c) and are symmetrical U-shaped or inverted U-shaped curves.
- 5If the 'a' value in y = ax² + bx + c is positive (a > 0), the parabola opens upwards; if 'a' is negative (a < 0), it opens downwards.
- 6Always label your axes (x and y) and include a scale on your Cartesian plane.
Worked examples
Example 1
Graph the linear equation y = -2x + 3.
Answer
A straight line passing through (0, 3), (1, 1), and (2, -1).
Using a table of values is also a valid method: e.g., for x=-1, y=5; for x=0, y=3; for x=1, y=1; for x=2, y=-1.
Example 2
Solve the following simultaneous equations graphically: y = x + 1 and y = -x + 5.
Answer
The solution is x = 2, y = 3 (or the point (2, 3)).
Always check your solution by substituting the x and y values back into both original equations.
Example 3
Sketch the graph of the parabola y = x² - 4.
Answer
A parabola opening upwards, with vertex at (0, -4), y-intercept at (0, -4), and x-intercepts at (-2, 0) and (2, 0).
Notice the symmetry around the y-axis (x=0). The vertex is the lowest point for this parabola.
Common mistakes
- ✗Not using a ruler for linear graphs, leading to inaccurate lines and solutions for simultaneous equations.
- ✗Incorrectly calculating y-values when creating a table of values, especially with negative numbers or squaring.
- ✗Confusing positive and negative gradients (e.g., drawing y = -2x + 3 as an increasing line).
- ✗Drawing parabolas with sharp corners instead of smooth curves, particularly at the vertex.
- ✗Not extending graphs sufficiently to show the full relationship or the point of intersection.
Exam tips
- ★Always use a sharp pencil and a ruler for drawing graphs to ensure accuracy.
- ★For simultaneous equations, plot at least three points for each line to ensure accuracy and identify the intersection clearly.
- ★When sketching parabolas, ensure you include points on both sides of the axis of symmetry and clearly show the vertex and intercepts.
- ★Check your graphical solutions by substituting the coordinates of the intersection point back into the original equations to verify they hold true.
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