--- title: "Recognising cubic, reciprocal and exponential graphs" description: "Describe the shapes and key features of y = x³, y = 1/x and y = kˣ, calculate their values, and match fixed-multiplier situations to exponential models." canonical: https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9 source: https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Recognising cubic, reciprocal and exponential graphs Describe the shapes and key features of y = x³, y = 1/x and y = kˣ, calculate their values, and match fixed-multiplier situations to exponential models. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9 ## Ready when they can - Describe cubic and reciprocal graph features, calculate points, and explain the reciprocal's excluded input and asymptotes - Identify a fixed multiplicative factor as exponential change, and calculate or compare values over stated steps - Apply a stated shift or reflection to one of these shapes ## Lesson: Recognising cubic, reciprocal and exponential graphs The shifts, stretches and reflections you have learned also work on cubic, reciprocal and exponential graphs. Plot (x, y) from the origin: move horizontally to x, then vertically to y, using signed coordinates. Choose a labelled scale for each axis. Study each shape's points, symmetry and behaviour near the axes to sketch a smooth curve and match it to a rule or situation. Adding a constant shifts vertically; replacing x with x - a shifts horizontally. *(drawing: The cubic passes through the origin. A half turn about the origin maps it onto itself.)* For y = x³, use three equal factors: x times x times x, with two multiplications. The curve increases through the origin and has a flat spot there. It has half-turn symmetry: (2, 8) pairs with (-2, -8). It is steep far from zero on either side and flatter near zero. On the negative side, it becomes less steep as x approaches zero; on the positive side, it becomes steeper as x grows. Recover x by finding the number whose cube is y. Check by cubing. For y = 1/x, x cannot be zero and y never equals zero. Its branches lie in the first and third quadrants. Near zero, y increases without bound from the right and decreases without bound from the left. Far from zero in either direction, y approaches zero. The axes are asymptotes. Doubling a nonzero input halves its output. For 1/(x - a), the vertical asymptote is x = a, where the denominator is zero. Recover x from nonzero y by solving xy = 1: x = 1/y. For y = 2^x, each increase of 1 in x doubles y. The curve is positive, passes through (0, 1), approaches zero to the left and rises faster to the right. A negative power reverses doubling: 2^-2 = 1/4. For y = k^x, k > 1 gives growth and 0 < k < 1 gives decay; both pass through (0, 1). A fixed positive starting amount multiplied by the same factor each step follows an exponential model. **Example.** To sketch y = x³, cube several inputs: -2, -1, 0, 1 and 2 give -8, -1, 0, 1 and 8. Plot the points and draw a smooth S-curve through the origin, flat there and steeper farther away. Check half-turn symmetry: (2, 8) pairs with (-2, -8). The points (3, 27) and (-3, -27) show how steep the tails become. **Example.** To sketch y = 1/x, use points (1, 1), (2, 1/2), (4, 1/4) and (8, 1/8). Larger positive inputs give outputs closer to zero. Negative inputs give the other branch: (-1, -1) and (-2, -1/2). Near zero from the right, inputs 1, 1/2, 1/4 and 1/8 give outputs 1, 2, 4 and 8. Draw two separate smooth branches approaching the axes, with no point at x = 0. Find a y-intercept by setting x = 0. For y = 4^x + 3, 4^0 = 1, so y = 4 and the intercept is (0, 4). A positive base raised to zero is 1. An added constant changes the output, including the intercept; a number inside the exponent changes the input instead. A fixed ratio identifies repeated multiplication: 2, 8, 32, 128 uses factor 4 each step. Adding 5% of a current amount B gives B + 0.05B = 1.05B, the same multiplier each year. To compare increases over a stated interval, evaluate both endpoints and subtract. For 3^x from x = 1 to 2, the increase is 9 - 3 = 6. Growing differences alone do not prove a model is exponential. Use the transformation rules on these functions too. Adding k to f(x) raises every output by k. Replacing x with x - h shifts corresponding points right h. A minus sign outside reflects in the x-axis: the new output is the negative of the old output. For y = -x³ at x = 10, first calculate 10³ = 1000, then negate it to get -1000. For y = 1/x + 3 at x = 4, find 1/4 = 0.25, then add 3 to get 3.25. **Recap.** The cubic is an increasing S-curve through the origin. The reciprocal has two branches approaching the axes. For k > 1, k^x is positive, approaches zero to the left and grows to the right. ## Practice 36 questions on this page, each with its working shown. ## Needs first - [Transformations of Graphs](https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M) ## Optional connections - [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf) ## Opens up - [Locating Roots and Iterative Methods](https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM) - [The exponential function and the number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ) - [Discharging a Capacitor and the Time Constant](https://lightmysky.com/learn/science/discharging-a-capacitor-and-the-time-constant-mt_5sjDPudyad) - [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc) - [Rational Functions, Asymptotes and Curve Sketching](https://lightmysky.com/learn/mathematics/rational-functions-asymptotes-and-curve-sketching-mt_U-f9YTNb2o) - [The Decay Constant and the Exponential Decay Law](https://lightmysky.com/learn/science/the-decay-constant-and-the-exponential-decay-law-mt_VIydSAys1W)