--- title: "Transformations of Graphs" description: "Describe and apply the effect of f(x) + a, f(x + a), -f(x) and af(x) on a graph, and give the coordinates of a marked point after the change." canonical: https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M source: https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M.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`. # Transformations of Graphs Describe and apply the effect of f(x) + a, f(x + a), -f(x) and af(x) on a graph, and give the coordinates of a marked point after the change. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M ## Ready when they can - Explain the direction of horizontal shifts such as f(x + 3), translate allowed-input domains, and identify what translations preserve or undo - Describe shifts, reflections and positive scale factors; track marked points, including turning points; compare transformed outputs and recover transformation parameters - Match transformed sketches or stated graph features to equations, and identify equivalent transformed equations ## Lesson: Transformations of Graphs In the last stop, swapping a point's coordinates reflected an inverse graph in y = x. Here you track points through translations, scaling and reflections in the coordinate axes. A point on y = f(x) has coordinates (u, v), where v = f(u). Read the new rule to find its image. These moves apply to many different graph shapes. *(drawing: A vertical translation changes y and leaves x unchanged.)* For f(x) + a, (u, v) becomes (u, v + a): positive a moves up; negative a moves down. For af(x), it becomes (u, av). Distances from the x-axis scale by |a|, the magnitude of a. Above 1 stretches; between 0 and 1 compresses; 1 leaves distances unchanged. Negative a also reflects in the x-axis, reversing highest and lowest points. An output of 0 stays 0. At a = 0, all defined outputs become 0; excluded inputs stay excluded. For f(x + a), solve x + a = u to keep the old output v. The image is (u - a, v). Positive a moves left; negative a moves right; zero leaves points unchanged. Thus f(x - a) moves right by a when a is positive. Horizontal translations preserve y-values, highest and lowest values, and horizontal distances. Every allowed input moves by the same amount. Equal opposite translations cancel: subtract a from x, then add a back. **Example.** Suppose f has its lowest point at (2, -5). Under f(x) + 2, it becomes (2, -3) and remains the lowest point. Under -f(x), it becomes (2, 5) and is the highest point. Translations preserve the graph's shape. Reflections reverse its orientation, while keeping distances unchanged. **Example.** Suppose f passes through (4, -4) and has its highest point at (1, -1). Under -2f(x), these become (4, 8) and (1, 2). The highest point becomes the lowest. Distances from the x-axis double, and the graph reflects in that axis. Adding a constant afterwards shifts every image by the same amount, so the new lowest point remains the lowest. **Example.** Combine moves by tracking each coordinate. Under Af(x - h) + k, (u, v) becomes (u + h, Av + k). Multiply y before adding k. A turning point at (-3, 4) becomes (-3, 9) under 2f(x) + 1. But 2(f(x) + 1) gives y = 2(4 + 1) = 10. Expanding those brackets gives 2f(x) + 2, so the two rules differ. **Example.** For f(ax), with a nonzero, solve ax = u. The image of (u, v) is (u/a, v). Distances from the y-axis scale by 1/|a|. Negative a also reflects in the y-axis. For example, (6, 9) becomes (2, 9) under f(3x). Under f(-x), it becomes (-6, 9). At a = 0, the new rule is the constant f(0) if that output exists; otherwise it has no allowed inputs. **Example.** Recover a point by undoing coordinate changes when scaling is nonzero. Under 3f(x - 2) - 4, image (1, 11) came from (-1, 5): x = 1 - 2 and y = (11 + 4)/3. Zero output scaling loses the original y. To find A in Af(x), solve v' = Av; divide by v when v != 0. If (2, 6) becomes (2, 20) under 3f(x) + k, solve 18 + k = 20 to get k = 2. **Example.** Substitute the rule for f. If f(x) = x^2, then Af(x - h) + k = A(x - h)^2 + k. When A != 0, its turning point is (h, k). One unit from x = h, its y-value is k + A, which helps identify A. When A = 0, the graph is the constant y = k and has no turning point. For f(x) = x^3, substitution gives A(x - h)^3 + k. Track points by the coordinate rule even when the graph has no turning point. **Example.** Absolute value, |x|, is the distance of x from zero. It gives a V-shaped graph with vertex (0, 0). Transform its points using the same coordinate rules. Also, |cv| = |c||v|. For example, 2|x + 1| = |2x + 2|. These two equations describe the same graph, even though one puts the multiplier outside and the other puts it inside. **Example.** A domain is the set of allowed inputs. A horizontal translation moves both ends of a restricted domain. If f is defined for -2 <= x <= 6, then f(x + 1) needs -2 <= x + 1 <= 6. Subtract 1 throughout: -3 <= x <= 5. The outputs stay the same. A vertical translation leaves the domain unchanged. **Example.** Compare two vertical translations at the same allowed input. If g(x) = f(x) + r and h(x) = f(x) + s, then g(x) - h(x) = r - s: the common f(x) cancels. For example, f(x) + 8 is always 7 units above f(x) + 1. You do not need to know the original output. **Recap.** Track each coordinate. Added outputs shift y; added inputs shift x in the opposite direction. Scaling uses positive magnitudes, with reflection separate. Read brackets before combining moves. *The lesson, practice questions and flashcards were written for this dataset.* ## Practice 36 questions on this page, each with its working shown. ## Needs first - [Inverse functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4) - [Function notation and evaluating functions](https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z) ## Opens up - [Modulus Functions and Modulus Equations](https://lightmysky.com/learn/mathematics/modulus-functions-and-modulus-equations-mt__W-aDpT5jz) - [Matrix multiplication and what it represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr) - [Recognising cubic, reciprocal and exponential graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9) - [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-)