What changes when I shift a function graph left, right, up, or down?
Quick answer
For a graph written as g(x)=f(x-h)+k, the graph of f moves right h units when h is positive, and up k units when k is positive; negative values move it left or down. The horizontal sign can feel backward: f(x+3) moves left 3, while f(x-3) moves right 3. A point (x,y) on f becomes (x+h,y+k) on g.
The question, the details, the next step.
I know a function’s graph can move, but I keep mixing up which signs move it which way. What should I look at first?
Separate the changes inside and outside the function. An added constant outside, as in f(x)+k, changes the output and shifts the graph vertically: positive k means up, negative k means down. A change to the input shifts it horizontally, with the direction opposite the sign you see inside: f(x+a) moves left a units if a is positive, while f(x-a) moves right a units. [S1]
A handy general form is g(x)=f(x-h)+k. Read it as “right h, up k,” allowing negative values: h=-2 means left 2, and k=-3 means down 3. The horizontal sign seems backward because it changes which input of the original function produces each output.
Can you show how that works with actual points, not just the formula?
Assume the original function is f(x)=x², and make the shifted function g(x)=f(x-3)+2. The minus 3 inside moves the graph right 3; the plus 2 outside moves it up 2. So every original point (x,y) becomes (x+3,y+2).
For example, f has these points:
- (-1,1) becomes (2,3).
- (0,0) becomes (3,2).
- (2,4) becomes (5,6).
You can check one: the original point (0,0) moves to (3,2), and g(3)=f(3-3)+2=f(0)+2=2. The new graph is the same shape and orientation, just translated. [S1]
What if the expression has a plus sign inside, or the shifts go in opposite directions?
Use the general form rather than guessing from the sign alone. For example, g(x)=f(x+4)-1 can be rewritten as f(x-(-4))-1, so it moves left 4 and down 1. Each point (x,y) becomes (x-4,y-1).
When shifts oppose each other, apply both: one changes the x-coordinate and the other changes the y-coordinate. For a check, if (1,5) is on f, then it becomes (-3,4) on this g. The function inside is evaluated at x+4, so the new input -3 uses the old input 1: -3+4=1.
Is there a quick method I can reuse on homework without drawing the whole graph?
Yes. First identify the horizontal change inside f and translate it into a point movement; then identify the outside vertical change. If the formula matches f(x-h)+k, move every marked point by (h,k). If it is written with a plus inside, treat that as a negative h. Finally, test one known point by substituting its new x-coordinate into the shifted formula.
These rules describe translations. They do not by themselves cover other transformations, such as changing a graph’s scale or reflecting it, so check whether the formula includes anything beyond input or output additions. [S1]
A practical checklist
Write the shifted function in the form g(x)=f(x-h)+k when possible.
Read the horizontal movement from h: positive is right; negative is left.
Read the vertical movement from k: positive is up; negative is down.
Move each known point (x,y) to (x+h,y+k).
Substitute one new x-coordinate to check that its output matches the moved point.
A few more useful answers
Why does f(x+4) move left instead of right?
The new graph uses the old function value at x+4. To get the old point at input 0, the new input must be -4, so that point appears 4 units to the left.
Do horizontal and vertical shifts change the graph’s shape?
A translation alone moves the graph without changing its shape or orientation. Other operations in the formula may cause different transformations.
How can I check the direction if I forget the sign rule?
Choose a known point on the original graph and ask which new input makes the shifted function use that old input. Then add the outside constant to its output.
Sources
References used for this guide.
- [S1] 3.5 Transformation of Functions — Algebra and Trigonometry 2e, OpenStax openstax.org · Checked