Exponential Function

y = a^x multiplies its value by a each time x rises by 1. That is why money earning interest or a splitting cell looks slow at first and then explodes upward. Above base 1 the curve soars; below 1 it fades toward 0, yet it always passes through (0, 1) with the x-axis as its asymptote. Here you set the base to reshape the curve, then slide the graph to watch the asymptote follow.

Shape — How the Base Changes the Curve

Base a2

💡 Every exponential passes through (0, 1)

①a^0 = 1 so the y-intercept is always 1

②Larger base → steeper rise to the right

③x → -∞ ⇒ y → 0 (x-axis is the asymptote)

Properties

Definition

y = a^x (a > 0, a ≠ 1)

Base a is a positive real, not equal to 1

📐 a > 1 vs 0 < a < 1

①a > 1: y grows fast (increasing)

②0 < a < 1: y → 0 as x grows (decreasing)

③Asymptote is the x-axis (y = 0)

④Range: y > 0 (always positive)

Translation of Exponential

x shift0

y shift0

Translation

y = a^x-p + q

Shift +p on x-axis, +q on y-axis → asymptote y = q

Exponential Equations & Inequalities

Exponential Equation

a^f(x) = a^g(x) ⟹ f(x) = g(x)

Same base → compare exponents

⚠️ Direction of Inequality Depends on the Base

①a > 1: a^m > a^n ⟺ m > n (inequality preserved)

②0 < a < 1: a^m > a^n ⟺ m < n (inequality flipped!)

③Always check whether the base is > 1 or < 1

Wrap-up

Exponential Core

y = a^x: domain = ℝ, range = (0, ∞), asymptote y = 0

Increasing when a > 1, decreasing when 0 < a < 1

🎯 Exam Points

①Passes through (0, 1): a^0 = 1

②a > 1 increases, 0 < a < 1 decreases

③Asymptote with translation: y = a^{x-p} + q → y = q

④Equations: equalize bases → compare exponents

⑤Inequalities: flip the sign when 0 < a < 1

Worked Examples & Past Exam

Example 1

Solve the exponential equation 2^x+1 = 8.

Rewrite 8 as a power of base 2. 8 = 2³.

2^x+1 = 2³

With equal bases, compare the exponents.

x + 1 = 3 ⟹ x = 2

▸ x = 2

Make both sides have the same base, then equate the exponents.

Example 2

Shift the graph of y = 2^x right by 3. Find the new equation and its asymptote.

A shift of +3 along the x-axis replaces x with x − 3.

y = 2^x-3

A horizontal shift does not change the asymptote.

asymptote: y = 0

▸ y = 2^x-3, asymptote y = 0

A horizontal shift keeps the asymptote; only a vertical shift moves it.

2022 KICE mock exam Math type, adapted

For 0 < a < 1, solve the exponential inequality a^2x < a^x+3.

①x > 3

②x < 3

③x > -3

④x < -3

⑤x > 1

▸ ① x > 3

Since the base is between 0 and 1, the function decreases and the inequality flips.

a^2x < a^x+3 ⟺ 2x > x + 3

Solve the inequality.

2x - x > 3 ⟹ x > 3

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Exponents & Logarithms

Logarithmic Function

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