Functions and Graphs - Estimating Gradients from Tangents

The variables \(x\) and \(y\) are connected by the equation \(y = 4x + \frac{20}{x} - 25.\)

Some corresponding values of \(x\) and \(y\) are given in the following table.

1. Find the value of \(h\) .
2. Using a scale of \(2\) cm to represent \(1\) unit, draw a horizontal axis for \(0 \le x \le 8\) .
   Using a scale of \(1\) cm to represent \(1\) unit, draw a vertical axis for \(-7 \le y \le 10\) .
   On your axes, plot the points given in the table and join them with a smooth curve.
3. Using your graph, find
   1. the minimum value of \(y\) .
   2. the solutions for the equation \(4x + \frac{20}{x} - 25 = -5\) .
4. By drawing a tangent, find the gradient of the curve at \(x = 2.5\) .

Graph-window correction: the minimum is slightly below \(-7\). Extend the vertical axis to at least \(-8\) to read it. The function is defined only for \(x\ne0\); use the positive-input branch shown in the table.