The exponential function y = aˣ multiplies its value by a every time x grows by 1.
That is why it describes changes that start slowly and then shoot up, like savings growing with compound interest or cells dividing in two again and again.
If the base is greater than 1 the graph soars to the right, and if it is between 0 and 1 it fades toward 0, but either way it passes through the point (0, 1) and has the x-axis as its asymptote.
In the first picture, changing the base changes how steep the curve is; in the second, shifting the graph moves the asymptote and the point (0, 1) along with it.
How the shape depends on the base
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The curve y = aˣ for the chosen base (gold) and the curves for bases 1.5, 2, 3 (faint)
2
💡 Every exponential graph passes through (0, 1)
①Since a⁰ = 1, the graph passes through (0, 1) whatever the base
②For x > 0 a bigger base climbs more steeply, and for x < 0 a bigger base hugs 0 more closely
③As x decreases, y gets as close to 0 as you like, so the x-axis is an asymptote
④The base in this picture is at least 1.2, so only increasing curves are drawn
Properties
Definition
y = ax (a > 0, a ≠ 1)
The base a is a positive number other than 1
📐 a > 1 versus 0 < a < 1
①If a > 1, y grows as x grows: an increasing function
②If 0 < a < 1, y shrinks as x grows: a decreasing function
③The graph of y = (1/a)ˣ is the graph of y = aˣ flipped over the y-axis
④Either way, the range is all positive numbers and the asymptote is the x-axis (y = 0)
Translation of Exponential
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The shifted curve (gold) of y = 2ˣ (faint), its asymptote (red dashes) and where the point (0, 1) has moved
0
0
Translation
y = ax-p + q
Shifting by p in the x-direction and q in the y-direction moves the asymptote to y = q. The base in the picture is 2
Exponential equations and inequalities
Exponential Equation
af(x) = ag(x) ⟹ f(x) = g(x)
Make the bases equal, then compare the exponents
⚠️ In an inequality, check first whether the base is above 1
①If a > 1, aᵐ > aⁿ ⟺ m > n: the inequality keeps its direction
②If 0 < a < 1, aᵐ > aⁿ ⟺ m < n: the inequality flips
③On a decreasing function, a larger exponent gives a smaller value
Summary
Exponential Core
y = ax: domain = ℝ, range = (0, ∞), asymptote y = 0
Increasing when the base is above 1, decreasing when it is between 0 and 1
🎯 Exam Points
①The graph always passes through (0, 1), since a⁰ = 1
②Increasing if a > 1, decreasing if 0 < a < 1
③The asymptote of y = aˣ⁻ᵖ + q is y = q
④Exponential equations: match the bases, then compare exponents
⑤Exponential inequalities: flip the inequality when the base is less than 1
Worked Examples & Past Exam
Example 1
Solve the exponential equation 2x+1 = 8.
1
Rewrite 8 as a power of 2: 2³.
2x+1 = 23
2
y = 2ˣ is increasing, so equal values mean equal exponents.
x + 1 = 3 ⟹ x = 2
▸ x = 2
With the same base (not 1), two different exponents can never give the same value. That is why matching the bases turns the equation into a simple comparison of exponents.
Example 2
Shift the graph of y = 2x right by 3. Find the new equation and its asymptote.
1
To move the graph 3 units in the x-direction, put x − 3 in place of x.
y = 2x-3
2
The graph only slid sideways, so the asymptote y = 0 stays where it is.
asymptote: y = 0
▸ y = 2x-3, asymptote y = 0
The asymptote is a horizontal line, so sliding it left or right leaves the same line. Only the vertical shift q moves the asymptote.
2022 KICE mock exam Math type, adapted
For 0 < a < 1, solve the exponential inequality a2x < ax+3.
①x > 3
②x < 3
③x > -3
④x < -3
⑤x > 1
▸ ① x > 3
1
For 0 < a < 1, y = aˣ is decreasing, so the order of the values is the reverse of the order of the exponents.