https://file.aiquickdraw.com/mathbot/file/17271758554396j03y3iq.jpg

To estimate the gradient of the graph at \( x = 3 \), we need to draw a tangent line to the curve at this point and then calculate the slope of this tangent line.

### Step 1: Drawing the Tangent Line
At \( x = 3 \), the \( y \)-value appears to be around 4 from the graph. We draw a tangent line that just touches the curve at this point without crossing it.

### Step 2: Estimating Points on the Tangent Line
To find the gradient, we need two points on the tangent line. Let's estimate two points near \( x = 3 \):

1. The tangent seems to pass through the point \( (2, 3.5) \).
2. The tangent also passes through approximately \( (4, 4.2) \).

### Step 3: Calculating the Gradient
The gradient (slope) of a line can be found using the formula:

\[
\text{Gradient} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the estimated points:

\[
\text{Gradient} = \frac{4.2 - 3.5}{4 - 2} = \frac{0.7}{2} = 0.35
\]

### Conclusion
The estimated gradient of the graph at \( x = 3 \) is approximately **0.35**.

Would you like further clarification on any part of this solution, or do you have any other questions?

### Related Questions:
1. How would you find the exact gradient of a curve analytically?
2. What is the significance of the gradient of a curve in real-world applications?
3. How does the method of drawing a tangent line relate to finding derivatives in calculus?
4. What are some common mistakes to avoid when estimating gradients graphically?
5. How can you improve the accuracy of gradient estimation on a graph?

**Tip:** When estimating gradients on graphs, try to be as precise as possible in selecting points and drawing the tangent line; small differences can affect the accuracy of your result!

Work out an estimate of the gradient of the graph when x = 3. You must show your working.

This problem demonstrates how to estimate the gradient of a graph at a specific point (x = 3) using tangent lines. The solution includes the estimation of two points near the tangent and the calculation of the gradient using the slope formula. Suitable for students in Grades 9-12.

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