8. (a) Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

To sketch the graph of the function $y = 3^x$, we will first identify key characteristics and points:

1. Intercept with the y-axis: When $x = 0$,

   $y = 3^0 = 1$.

   Therefore, the graph crosses the y-axis at the point $(0, 1)$.

2. Intercept with the x-axis: For $y = 3^x$ to equal 0,

   there are no points where this is true since an exponential function never crosses the x-axis.

3. Behavior of the graph:

   • As $x \to -\infty$, $y \to 0$.
   • As $x \to \infty$, $y \to \infty$.

4. Shape of the curve: The graph is increasing for all $x \in \mathbb{R}$, has no turning points, and approaches the x-axis but never touches it.

In summary, the graph passes through the point $(0, 1)$ and approaches the x-axis as $x$ decreases, indicating that it does not touch the x-axis.