Know and use the function a^x and its graph, where a is positive. Know and…

The idea

An exponential function has the variable in the exponent: y = a^x with a > 0. Its graph always passes through (0, 1) because a⁰ = 1, stays entirely above the x-axis since a^x > 0 for all real x, and has the x-axis as a horizontal asymptote. If a > 1 the graph increases; if 0 < a < 1 it decreases, and the two cases are reflections of each other in the y-axis. The special base e ≈ 2.718 is defined so that the curve y = e^x has gradient equal to its own value at every point. To sketch, plot (0, 1), (1, a) and a second guide point such as (−1, 1/a), then draw a smooth curve approaching but never touching the axis. Recognise transformations: y = a^(x + c) shifts left by c, y = a^x + d shifts up by d, and y = a^(kx) stretches horizontally.

What earns the marks

• States that a^x is defined for all real x provided a > 0, and that a^x is always positive, so the graph lies wholly above the x-axis.
• Identifies the y-intercept (0, 1) for every valid base and explains it follows from a⁰ = 1.
• Distinguishes increasing behaviour when a > 1 from decreasing behaviour when 0 < a < 1, linking this to the size of the base.
• Describes the x-axis as a horizontal asymptote and explains that the curve approaches it as x → −∞ (for a > 1) without meeting it.
• Sketches y = e^x with correct shape, intercept and asymptote, and states e ≈ 2.718 as the base making the gradient equal to the function value.
• Applies transformations correctly, for example reading y = e^(x − 2) + 3 as a translation of y = e^x by 2 right and 3 up.

Common mistakes and how to put them right

• Treating a^x as a polynomial and drawing a parabola or straight line.

  Put it right by tabulating values such as a⁰, a¹ and a^(−1) and plotting a smooth curve with an asymptote.

• Believing the graph can cross or touch the x-axis for some x.

  Put it right by noting a^x > 0 for every real x when a > 0.

• Assuming every exponential graph increases.

  Put it right by checking whether the base lies above or below 1, since 0 < a < 1 gives a decreasing curve.

• Confusing the roles of base and exponent, for example reading 2^x as x².

  Put it right by stressing that in a^x the variable is the exponent.

Exam tips

• Always mark the intercept (0, 1) and the asymptote on any exponential sketch, as these features carry the meaning of the curve.
• When a question gives a point on y = a^x, substitute the coordinates and solve for a, checking that your value is positive.
• For transformed exponentials, identify the base curve first, then apply each translation or stretch in turn to avoid combining them incorrectly.
• Use a quick table of three or four values to confirm the shape before drawing, especially when the base is a fraction.