F: Exponentials and logarithms: Know and use the function a^x and its graph, where a is positive. Know and use the function e^x and its graph.

The idea

For a positive base a, a^x is defined for all real x and is always positive. Its graph passes through (0,1) and lies above the x-axis. When a>1 the graph increases and the x-axis (y=0) is a horizontal asymptote as x→-∞; when 0<a<1 it decreases and y=0 is an asymptote as x→+∞. The case a=1 is the constant function y=1, a horizontal line with no asymptote, so the asymptote statement applies only to a≠1. The base e≈2.718 is defined so that the gradient of e^x at x=0 is exactly 1, giving the self-derivative property. To sketch y=a^x, plot (0,1), (1,a) and (-1,1/a), then draw a smooth curve approaching but never touching the x-axis. Transformations: y=a^(x+b) shifts left by b, y=ka^x stretches vertically by k, and y=a^(-x) reflects in the y-axis.

What earns the marks

• State that for a>0, a^x is defined for all real x and is always positive.
• Identify the y-intercept as (0,1) for any positive base a.
• Describe the horizontal asymptote y=0 and the increasing/decreasing behaviour for a>1 and 0<a<1, and state that a=1 gives the constant function y=1 with no asymptote.
• Sketch y=e^x showing correct shape, intercept (0,1) and asymptotic behaviour as x→-∞.
• Apply transformations such as y=e^(x+2) or y=3e^x to the graph of y=e^x, working out key point coordinates.
• Use the graph to solve equations or inequalities involving a^x or e^x.

Common mistakes and how to put them right

• Thinking that a^x can be negative for some x.

  Put it right by noting that a positive base raised to any real power is positive.

• Drawing the graph crossing the x-axis.

  Put it right by showing the x-axis as an asymptote that the curve approaches but never meets (for a≠1).

• Treating a=1 as having the same asymptote as other bases.

  Put it right by noting y=1^x=1 is the constant line y=1, with no asymptote.

• Forgetting that e is a specific constant approximately 2.718, not a variable.

  Put it right by treating e^x as a particular exponential function.

Exam tips

• When sketching, mark the intercept (0,1) and, for a≠1, the asymptote y=0 to gain clarity and method marks.
• For transformations, work out the coordinates of key points such as the intercept and one other point before drawing the curve.
• If asked to compare a^x and e^x, comment on the base value and the resulting steepness rather than just repeating the shape.
• Use the graph to justify solutions to equations; a rough sketch can confirm the number of intersections.