A reciprocal graph y = k/x has two curved branches that never touch the x-axis or y-axis, sitting in opposite quadrants depending on the sign of k. An exponential graph y = k·aˣ is a single smooth curve that crosses the y-axis at y = k and flattens towards y = 0 on one side without ever reaching it. Both graphs are recognised first by where x sits in the equation: in the denominator for reciprocal, in the exponent for exponential.
Both graphs curve smoothly with an asymptote, so at a glance they can look similar. Students who skip checking where x actually sits in the equation end up sketching the wrong shape: one continuous exponential curve for a reciprocal function, a missing second branch, or the wrong number of asymptotes altogether.
The fix: look at the equation before you touch your pencil. x on the bottom of a fraction means reciprocal. x up in the power means exponential. Everything else about the sketch follows from that one check, plus the sign of the constants involved.
Step 1
Identify: x in denominator (reciprocal) or x in exponent (exponential)
Step 2
Reciprocal: check sign of k for quadrants; mark x = 0 and y = 0 as asymptotes
Step 3
Exponential: find y-intercept at x = 0; mark y = 0 as the one asymptote
Step 4
Plot 2 guide points each side, sketch curve approaching asymptote(s)
A reciprocal graph has the general form y = k/x, where k is a non-zero constant. Since x cannot equal 0 (division by zero is undefined) and y can never equal 0 either, the graph has two asymptotes: the vertical line x = 0 and the horizontal line y = 0. The curve consists of two separate branches, and it never crosses either axis.
Worked example 1: sketch y = 6/x
k = 6, which is positive, so the curve lies in the quadrants where x and y have the same sign: the first quadrant (both positive) and the third quadrant (both negative).x = 1, y = 6. When x = 2, y = 3. When x = 3, y = 2. When x = 6, y = 1.x: when x = −1, y = −6; when x = −2, y = −3; when x = −6, y = −1.x increases, and a mirror-image branch in the third quadrant. Neither branch touches x = 0 or y = 0.Worked example 2: sketch y = −4/x
k = −4, which is negative, so the curve lies where x and y have opposite signs: the second quadrant (x negative, y positive) and the fourth quadrant (x positive, y negative).x = 1, y = −4. When x = 2, y = −2. When x = 4, y = −1. When x = −1, y = 4. When x = −2, y = 2. When x = −4, y = 1.Quick check: the sign of k tells you the pair of quadrants instantly: positive k means quadrants 1 and 3, negative k means quadrants 2 and 4. You do not need to plot a single point to know which two quadrants the branches sit in.
An exponential graph has the general form y = k·aˣ, where k is a non-zero constant (positive in almost every O-Level question) and a is a positive constant not equal to 1. Unlike the reciprocal graph, this is one continuous curve. It crosses the y-axis at y = k (since a⁰ = 1), and it has exactly one asymptote: y = 0. The curve never touches the x-axis, no matter how large or small x becomes.
The value of a decides which way the curve runs. If a > 1, it rises steeply to the right and flattens towards y = 0 on the left. If a is between 0 and 1, it does the opposite: high on the left, flattening towards y = 0 on the right. The two worked examples below are one of each.
Worked example 3: sketch y = 2ˣ
x = 0, y = 2⁰ = 1. The curve crosses the y-axis at (0, 1).x = 1, y = 2. x = 2, y = 4. x = 3, y = 8. The curve rises quickly as x increases.x = −1, y = ½. x = −2, y = ¼. x = −3, y = ⅛. As x decreases, y gets closer and closer to 0 but never reaches it.y = 0) on the left, never crossing it.Worked example 4: sketch y = 3(½)ˣ
k = 3 and a = ½. At x = 0, y = 3 × 1 = 3, so the curve crosses the y-axis at (0, 3).a is between 0 and 1, the curve decays as x increases. x = 1, y = 1.5. x = 2, y = 0.75. For negative x it rises instead: x = −1, y = 6. x = −2, y = 12.(0, 3), flattening towards y = 0 on the right without ever touching it.The step students get wrong
Drawing the exponential curve as if it eventually reaches y = 0 and stops, or crosses over to negative y-values. It never does either. An exponential graph of the form y = k·aˣ with k positive stays above the x-axis for every value of x, getting closer and closer to 0 without ever touching it. Also common: forgetting that a reciprocal graph has two branches and two asymptotes, and sketching it as one continuous curve like an exponential graph instead.
Exam tip: if a question gives you a graph and asks you to identify the equation, count the branches first. Two separate curves that avoid both axes means reciprocal. One smooth curve crossing the y-axis and flattening on one side only means exponential. The same shape-first habit pays off when you are solving equations graphically.
Can a reciprocal graph have a y-intercept?
No. Since x = 0 is not allowed in y = k/x (division by zero is undefined), the graph never crosses the y-axis. This is different from the exponential graph, which always crosses the y-axis at y = k.
What happens to y = k·aˣ if k is negative?
The whole curve reflects below the x-axis. It still has the asymptote y = 0, and it still crosses the y-axis at y = k, but the curve now sits entirely below the x-axis instead of above it. Sketch the positive-k curve first, then flip it over the x-axis.
How is this connected to index laws?
Every guide point you calculate on an exponential graph is an application of the index laws, especially the rule that a negative power gives a reciprocal (for example 2⁻¹ = 1/2) and that any non-zero number to the power 0 equals 1. If your index laws are shaky, the exponential guide-point calculations will go wrong even if you understand the shape correctly.
Are these graphs tested the same way as other O-Level graph sketches?
Yes. As with the wider set of y = axⁿ graph sketches, you are usually asked to sketch the shape correctly with key features labelled (intercept, asymptotes, general direction), rather than to plot every point with graph paper precision. Get the shape, the intercept, and the asymptotes right and you earn the marks.
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