Math Problem Statement

What is the gradient of the line shown below? Give your answer as an integer or as a fraction in its simplest form.

Solution

To find the gradient (slope) of the line, we use the formula:

slope(m)=ΔyΔx=change in ychange in x\text{slope} (m) = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y}{\text{change in } x}

From the graph, we can identify two points on the line to calculate the change in yy and xx.

  • Point 1: (0,2)(0, 2) (where the line crosses the y-axis)
  • Point 2: (2,6)(2, 6)

Now, calculate the changes: Δy=62=4\Delta y = 6 - 2 = 4 Δx=20=2\Delta x = 2 - 0 = 2

Thus, the gradient is: m=42=2m = \frac{4}{2} = 2

So, the gradient of the line is 2.

Would you like more details or have any questions? Here are five related questions to expand on this concept:

  1. What does the gradient represent in terms of real-world applications?
  2. How would the equation of the line look in slope-intercept form?
  3. What happens to the gradient if the line becomes horizontal or vertical?
  4. How can you determine the gradient if the points are not clear from a graph?
  5. What is the difference between positive and negative gradients?

Tip: The steeper the line, the larger the absolute value of the gradient.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Coordinate Geometry

Formulas

Gradient formula: m = (change in y) / (change in x)

Theorems

Slope of a straight line

Suitable Grade Level

Grades 7-9