To find the gradient (slope) of a straight line on a graph, you usually need at least two points on the line. However, if you have one point, like the y-intercept, and you know the gradient is 3, there must be a way to infer this information directly from the graph or the equation of the line.

Understanding the Gradient of a Line

The gradient (or slope) of a line tells us how steep the line is. It's usually defined as the "rise over run," which is the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change) between two points on the line:

$\text{Gradient (m)} = \frac{\Delta y}{\Delta x}$

If the gradient is given as 3, it means that for every 1 unit you move to the right (along the x-axis), the line goes up by 3 units (along the y-axis).

How to Find the Gradient When Only Given One Point

If you see only the y-intercept (the point where the line crosses the y-axis), there might be some visual clues or numerical hints on the graph that help you deduce the gradient:

1. Visual Slope Estimate: If the graph is marked with a grid, you can count the number of units the line rises vertically for a given horizontal run. For example, if you start at the y-intercept, look at the line's slope and see if it rises 3 units for every 1 unit of horizontal movement.

2. Equation of the Line: If the line's equation is given in slope-intercept form, $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept. If you know the y-intercept and the equation, you can directly read the gradient from it. If the gradient is 3, then the equation would look like $y = 3x + c$.

Conclusion

If your graph doesn’t show any other points but the y-intercept, you might need to use clues like the equation of the line or visual indications on the graph (like how the line is drawn relative to the grid). Since the answer is given as 3, that suggests that either:

• You can visually determine that the line rises 3 units for every 1 unit it moves horizontally.
• There is an equation provided that shows the gradient as 3.

Would you like to explore how to identify the gradient from a different example or learn more about related concepts?

Follow-Up Questions:

1. How do you find the gradient if you have two points on the line?
2. What is the slope-intercept form of a linear equation?
3. How does the gradient relate to parallel and perpendicular lines?
4. Can you determine the gradient from a line’s angle with the x-axis?
5. How does the gradient change when the line is horizontal or vertical?

Tip: When looking at a graph, always check for any grid lines or numerical scales, as they can help you estimate the gradient visually.