Graphs Of Functions
Reciprocal and exponential graphs
Reciprocal graphs — \( y = k/x \)
The reciprocal function \( y = \dfrac{k}{x} \) produces a curve called a hyperbola. It is defined for all values of \( x \) except \( x = 0 \).
Features of \( y = k/x \)
- Two asymptotes: \( x = 0 \) (y-axis) and \( y = 0 \) (x-axis). Draw these as dashed lines.
- Sign of \( k \):
- \( k > 0 \): branches in Quadrant 1 (top-right) and Quadrant 3 (bottom-left).
- \( k < 0 \): branches in Quadrant 2 (top-left) and Quadrant 4 (bottom-right).
- No intercepts: the curve never crosses either axis.
- Symmetry: the curve has two lines of symmetry: \( y = x \) and \( y = -x \).
- As \( x \to 0 \), \( y \to \pm\infty \). As \( x \to \pm\infty \), \( y \to 0 \).
Reciprocal squared — \( y = k/x^2 \)
- Since \( x^2 \) is always positive, \( y \) has the same sign as \( k \) for all \( x \neq 0 \).
- \( k > 0 \): both branches are above the x-axis (Q1 and Q2).
- \( k < 0 \): both branches are below the x-axis.
- Asymptotes are still \( x = 0 \) and \( y = 0 \).
- The curve is symmetric about the y-axis (even function).
\( y = k/x \) vs \( y = k/x^2 \): For \( y = k/x \) the two branches are in opposite quadrants (one positive, one negative region). For \( y = k/x^2 \) both branches are on the same side of the x-axis. This is the most-tested distinction.
Exponential graphs — \( y = ab^x \)
An exponential function has the form \( y = ab^x \) where \( a > 0 \) and \( b > 0,\; b \neq 1 \). The variable is the exponent (power), not the base.
Features of \( y = ab^x \)
- y-intercept: When \( x = 0 \), \( y = ab^0 = a \cdot 1 = a \). So the curve always passes through \( (0, a) \).
- Growth or decay?
- \( b > 1 \): exponential growth — curve rises steeply to the right.
- \( 0 < b < 1 \): exponential decay — curve falls and approaches zero from above.
- Horizontal asymptote: \( y = 0 \) (the x-axis). The curve approaches it but never reaches it (for \( a > 0 \)).
- No x-intercepts (when \( a > 0 \)).
- The curve is always positive when \( a > 0 \) and \( b > 0 \).
Worked example — using an exponential graph to solve an equation
The graph of \( y = 3^x \) is drawn. Use it to solve \( 3^x = 7 \).
Draw the horizontal line \( y = 7 \) on the graph. Read off the x-value at the intersection — this is the solution.
Using logarithms to check: \( x = \dfrac{\log 7}{\log 3} = \dfrac{0.845}{0.477} \approx 1.77 \).
The graph reading should give approximately \( x \approx 1.8 \) — within the acceptable tolerance for a graph answer (typically ±0.1).
Exponential growth and decay in context
Real-world exponential models:
| Context | Model | Notes |
|---|---|---|
| Population growth | \( P = P_0 \times r^t \), \( r > 1 \) | Grows without limit (in theory) |
| Radioactive decay | \( N = N_0 \times (0.5)^{t/t_{1/2}} \) | Halves every half-life \( t_{1/2} \) |
| Compound interest | \( A = P(1 + r)^n \) | Growth; \( r \) is the rate per period |
| Depreciation | \( V = V_0(1 - r)^n \) | Decay; \( r \) is rate of loss |
When a question says "show that the relationship is exponential," draw \( \log y \) against \( x \) — if the points form a straight line, the original data follows \( y = ab^x \). This is a linearisation technique used in Paper 2 graph questions.
Do not confuse \( y = x^2 \) (a quadratic — power of x) with \( y = 2^x \) (an exponential — x is the power). They have completely different shapes and growth rates.