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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
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.
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.
2 · Watch
3 · Play
Add 3 outside the bracket, then inside it, and watch the curve go two different ways.
Take it off screen
Where it sits
Learn first
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.