The two tangents are intersecting the and -axes, and they are also intersecting each other at the point labelled as .

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**Part A – Identifying the relationship between the slopes of the two tangents**

Use an algebraic approach to find the exact equation of the two tangents to the graphs of and , at .

Following a similar approach or technology, determine the exact equation of the two tangents at for 2, 3, 4.

Using results obtained, complete the table below relating the slopes of those two tangents.

Value of | Equation of the tangent to | Slope of the tangent to | Product of the two slopes | ||

1 | |||||

2 | |||||

3 | |||||

4 |

Describe any patterns observed in the table. Outline a conjecture for the equation of the tangent to these functions, and another conjecture for the product of the two slopes. Prove your conjectures.

**Part B – Investigating the relationship between the ** **-intercepts of the two tangents**

Determine the -intercepts of each set of equations of tangent identified in Part A.

Summarise your results in a table to investigate the products of the -intercepts for each set of equations.

Formulate a conjecture and prove it.

**Part C – Exploring further relationships between exponential functions and their tangents**

Following a similar process outlined in Part A and B above, explore the effect of transformation on and may have on the conjecture found previously.

Alternatively, you may like to consider various conjectures on exponential functions such as and or transformation of these functions.