Introduction to Linear Polynomials · Lesson 6 of 7
Visualizing Linear Relationships
“Put the relationship on a graph and the pattern stops playing hide-and-seek.”
• Plot a linear relationship as a straight line. • Verify whether a point lies on a given line. • Understand the slope a in y = ax + b. • Understand the y-intercept b. • Recognise positive slope, negative slope and parallel lines.
A linear relationship can be represented numerically by a table, algebraically by an equation, and geometrically by a straight line. The graph gives a visual picture of constant change.
Two Points Are Enough to Draw a Line
To graph y = 2x + 1, choose any two convenient x-values, calculate the corresponding y-values, plot the points and join them with a straight line.
| x | Calculation | y |
|---|---|---|
| 0 | 2(0)+1 | 1 |
| 3 | 2(3)+1 | 7 |
Problem
Does (7,15) lie on y = 2x + 1?
- 1.Substitute x = 7 and y = 15.
- 2.Right side = 2(7) + 1 = 15.
- 3.Left side is also 15.
- 4.Therefore (7,15) lies on the line.
Recognising the Equation from Points
If several plotted points share a simple relationship, the equation may be visible directly. For points (−1,−3), (0,0), (1,3), (3,9) and (4,12), every y-value is three times x, so the equation is y = 3x.
Similarly, for points (−3,6), (−2,4), (0,0), (1,−2), (2,−4), (3,−6), each y-value is −2 times x, so the equation is y = −2x.
Lines of the Form y = ax
Every line y = ax passes through the origin because when x = 0, y = 0. The value a controls both direction and steepness.
| Equation | Slope behaviour |
|---|---|
| y = 1/2 x | Positive, less steep than y = x |
| y = x | Positive reference line |
| y = 2x | Positive, steeper than y = x |
| y = −1/3 x | Negative, gentle downward slope |
| y = −x | Negative |
| y = −3x | Negative, steeper downward slope |
In a linear equation y = ax + b, a represents the constant rate of change and is called the slope of the line.
For positive a, the line rises from left to right and represents linear growth. For negative a, the line falls from left to right and represents linear decay.
What Happens When b Changes?
Now compare y = 2x − 1, y = 2x + 1 and y = 2x + 5. The coefficient of x is the same in all three equations, so the lines have the same slope. Only the constant term changes.
Because the slopes are equal, the lines are parallel. Changing b shifts the line up or down without changing its steepness.
The y-Intercept
Set x = 0 in y = ax + b. Then y = b. Therefore every line y = ax + b crosses the y-axis at (0,b). The number b is called the y-intercept.
The y-intercept is the value b in y = ax + b. It is the y-coordinate of the point where the line crosses the y-axis.
| Equation | y-intercept | Point on y-axis |
|---|---|---|
| y = 2x + 5 | 5 | (0,5) |
| y = x + 3 | 3 | (0,3) |
| y = 3x − 2 | −2 | (0,−2) |
Putting a and b Together
| Change in equation | Effect on graph |
|---|---|
| Change a, keep b fixed | Slope changes, y-intercept stays fixed |
| Keep a fixed, change b | Lines shift and remain parallel |
| a > 0 | Line rises left to right |
| a < 0 | Line falls left to right |
| b = 0 | Line passes through origin |
Practice Problems
- Draw y = 4x, y = 2x and y = x on the same axes. Compare their steepness.
- Draw y = −6x, y = −3x and y = −x. Compare their slopes.
- Draw y = 5x and y = −5x and describe the difference.
- Draw y = 3x − 1, y = 3x and y = 3x + 1. Explain why they are parallel.
- Draw y = −2x − 3, y = −2x and y = 2x + 3. Compare slope and y-intercept.
- For y = −3x + 4, identify slope and y-intercept and find the point where it cuts the y-axis.
- For 2y = 4x + 7, rewrite it as y = ax + b and identify a and b.
- For 5y = 6x − 10 and 3y = 6x − 11, identify slopes and decide whether any pair is parallel.
Key Takeaways
• The graph of a linear relationship is a straight line. • Two points determine a straight line. • A point lies on a line if its coordinates satisfy the equation. • In y = ax + b, a is the slope and b is the y-intercept. • Same slope with different y-intercepts gives parallel lines. • Positive slope represents growth and negative slope represents decay.
Next, we revise the chapter as a whole and practise moving between expressions, patterns, equations, tables and graphs.