11.2 Hyperbola, Exponential and Logarithmic Functions

Quick Answer: Identify the family first. A hyperbola y = a/(x − p) + q is missing one x and one y (the asymptotes). An exponential y = bˣ stays above the x-axis and has a horizontal asymptote. A logarithm y = log_b x lives to the right of a vertical asymptote. A cubic is a polynomial through all real x, listed with these families in the MAT booklet function list.

NBT Test Content on nbt.ac.za asks you to identify graphs of functions and to solve routine problems given verbally, algebraically, or graphically. Independent OpenExamPrep teaching below uses original equations. Other graphs such as circles can appear as well; a circle is a graph, but solving for y does not give a single function.

Hyperbolas

The basic rectangular hyperbola is y = k/x, or xy = k, with k ≠ 0.

• Vertical asymptote: x = 0 (the y-axis)
• Horizontal asymptote: y = 0 (the x-axis)
• Domain: x ≠ 0
• Range: y ≠ 0
• Intercepts: none, because x = 0 is forbidden and y is never 0

The useful school form is the shifted hyperbola

y = a/(x − p) + q

Worked example: y = 6/(x − 1) − 2

• Vertical asymptote x = 1, horizontal asymptote y = −2
• y-intercept: x = 0 gives 6/(−1) − 2 = −8, point (0, −8)
• x-intercept: 0 = 6/(x − 1) − 2 → 2 = 6/(x − 1) → x − 1 = 3 → x = 4, point (4, 0)

Signs of a: if a > 0, the two branches sit in the regions that match the unshifted y = k/x pattern relative to the new axes x = p and y = q. You do not need a calculator to plot two convenient points. For y = 6/(x − 1) − 2, try x = 2: y = 6 − 2 = 4. Try x = 4: y = 0, which we already have.

A frequent trap is to treat the horizontal asymptote as a y-intercept. The graph may cross the y-axis far from y = q. Another trap is writing domain as x ≥ p (that is a square-root story, not a hyperbola).

Exponential functions

An exponential function has the variable in the exponent: y = bˣ with b > 0, b ≠ 1. (y = 2 × 3ˣ is the same family, vertically stretched.)

Standard features of y = bˣ:

• Domain: all real x
• Range: y > 0
• y-intercept: (0, 1), because b⁰ = 1
• x-intercept: none — the graph never meets the x-axis
• Horizontal asymptote: y = 0, approached as x → −∞ if b > 1, or as x → +∞ if 0 < b < 1
• Vertical asymptote: none

If b > 1, the graph increases (growth). If 0 < b < 1, it decreases (decay). y = (1/2)ˣ is the reflection of y = 2ˣ in the y-axis.

Shifted form y = bˣ⁻ʰ + k:

• Horizontal asymptote becomes y = k
• Range becomes y > k if the stretch is positive
• y-intercept is b⁻ʰ + k, which is usually not 1

Example: y = 2ˣ − 4

• Horizontal asymptote y = −4
• y-intercept: 1 − 4 = −3
• x-intercept: 2ˣ = 4, so x = 2, point (2, 0)
• Range: y > −4

On MAT, options often claim that y = 3ˣ meets the x-axis, or that it is undefined for x < 0. Both are false. Negative inputs are legal: 3⁻² = 1/9.

Logarithmic functions

A logarithm is the inverse of an exponential. y = log_b x means bʸ = x, with b > 0, b ≠ 1, and x > 0.

Standard features of y = log_b x:

• Domain: x > 0
• Range: all real y
• x-intercept: (1, 0), because log_b 1 = 0
• y-intercept: none — x = 0 is not in the domain
• Vertical asymptote: x = 0
• Horizontal asymptote: none

Shifted form y = log_b (x − p) + q:

• Domain: x > p (argument strictly positive)
• Vertical asymptote: x = p
• x-intercept: log_b (x − p) + q = 0 → log_b (x − p) = −q → x − p = b⁻q

Example: y = log₃ (x − 4)

• Domain x > 4, not x ≥ 4 (log of 0 is undefined)
• Vertical asymptote x = 4
• x-intercept: log₃ (x − 4) = 0 → x − 4 = 1 → x = 5
• No y-intercept, because x = 0 is not greater than 4

Because exp and log are inverses, the graph of y = log_b x is the reflection of y = bˣ in the line y = x. That is why one family has a horizontal asymptote and the other has a vertical asymptote.

MAT will happily hide the argument: y = log₂ (2x − 6) needs 2x − 6 > 0, so x > 3. The vertical asymptote is 2x − 6 = 0, x = 3.

Cubics, as named in the MAT booklet function list

The booklet list of functions includes linear, quadratic, hyperbola, cubic, exponential and logarithmic. A cubic is a degree-3 polynomial, for example y = x³ − 4x, or y = 2(x + 1)³ − 3.

What MAT-style items typically want:

• Domain of a cubic polynomial: all real x. Range: all real y (a cubic always runs from −∞ to +∞).
• At least one x-intercept (odd degree). There may be one or three real roots, counting a repeated root as a touch-and-turn or a flattening at the axis.
• y-intercept = the constant term, f(0).
• Possible local max and min if the cubic is not strictly increasing, but no vertical or horizontal asymptotes. Do not give a cubic the missing-line look of a hyperbola.

Example: y = x³ − 4x = x(x − 2)(x + 2). Intercepts at −2, 0, and 2. Odd function: f(−x) = −f(x), so rotational symmetry about the origin. A shifted cube such as y = (x − 1)³ + 2 still has no asymptotes; (1, 2) is an inflection, not a hyperbola hole.

Circles, which the same booklet notes as other graphs, are not functions of x if you solve for y (two values). (x − 3)² + (y + 1)² = 16 is a circle, centre (3, −1), radius 4 — a graph you may still have to recognise, or use as a region boundary.

Matching a sketch to an equation

Work in this order on the no-calculator paper:

1. Asymptotes. Two perpendicular asymptotes, one vertical and one horizontal → hyperbola. Only a horizontal asymptote, graph never crosses it, defined for negative x → exponential. Only a vertical asymptote, defined on one side of it → logarithm. No asymptotes, wiggle through the plane → cubic (or quadratic if it is a single smile/frown).
2. Intercepts. Does it meet the y-axis? Exponential: yes. Log: usually no, unless shifted far enough that p < 0. Hyperbola: maybe.
3. A single test point. For y = 2ˣ, the point (1, 2) is cheap. For y = log₂ x, the point (8, 3) is cheap if you know powers of two.

Independent OpenExamPrep practice: name the family from the picture before you expand. That is the MAT skill — identification — not a calculator plot.