--- 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: 2026-09-02 --- > **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 - Say which way f(x + 3) moves the graph and why the shift looks backwards - Give the image of a marked turning point after a stated transformation - Match a transformed sketch to its equation ## Lesson: Transformations of Graphs The last stop reflected a graph in the line y = x to get the inverse. That was the first time a whole graph moved by a rule written in function language. This stop generalises it. Every point on the graph of f has coordinates (x, f(x)), and a transformation is a new rule built from f that moves every point at once. The skill: read the new rule, say which way each point goes, and give the coordinates of any marked point after the move. Four moves cover everything: up and down, left and right, flip, and stretch. *(drawing: One rule, applied to every point at once.)* Here are the moves. f(x) + a adds a to every output, so the graph slides up by a, and f(x) - a slides it down. -f(x) flips every output's sign, a reflection in the x-axis, turning the graph upside down. af(x) multiplies every output by a, a stretch by a factor of a parallel to the y-axis, and a factor between 0 and 1 squeezes the graph in. In all of these, the x-coordinate of every point stays exactly where it is. f(x + a) slides the graph left by a, and f(x - a) slides it right. The y never changes in a horizontal move. Why does f(x + a) go left? The move acts on the input. To get the output the graph had at x = 0, the new rule needs the input -a, so that output lands at x = -a: earlier. The direction looks backwards because the rule asks for the input sooner. And the reading trick: a number outside the brackets changes heights, and a number inside the brackets changes positions. Say which one you are looking at before you move a single point, and the move is decided. **Example.** Where does a marked point go? The graph of f has its lowest point at (2, -5). Where is the lowest point of f(x) + 2? The rule only changes heights: every y goes up by 2, every x stays put. So the lowest point moves from (2, -5) to (2, -3). Same point, different move: under -f(x) it lands at (2, 5), and the lowest point becomes the highest. The shape of the graph is untouched in both cases; only the labels on the points change. **Example.** Now a stretch and a flip together. The graph of f passes through (4, 4) and has a highest point at (1, -1). What does -2f(x) do to each? The rule multiplies every output by -2: the point (4, 4) lands at (4, -8), and the highest point (1, -1) lands at (1, 2). Notice what happened to the labels. Doubling by a negative flips the graph, so the highest point becomes the lowest, and every distance from the x-axis doubles. The x-coordinates, 4 and 1, are untouched. **Example.** The order matters when a stretch and a shift share one rule. The graph of f has a turning point at (-3, 4). Under 2f(x) + 1, the y goes from 4 to 2 times 4 plus 1, which is 9: the point lands at (-3, 9). But 2(f(x) + 1) means add 1 first, then double: 4 becomes 5, then 10. Same numbers, different brackets, one unit of difference. Read the brackets before you touch the point, and the order takes care of itself. **Recap.** Read the new rule in pieces: numbers outside the brackets change heights, so the graph slides, stretches or flips, while numbers inside the brackets change positions, and a marked point moves by applying each move to its coordinates. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Inverse Functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4) ## Opens up - [Recognising Cubic, Reciprocal and Exponential Graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9)