Shifts and Stretches

Calculus Readiness · Axiom Academy

Slide a graph up, down, left, or right without changing its shape — and meet the minus sign that pushes the opposite way. 1. Vertical Shifts: Add Outside Add a constant to the output and the whole graph slides straight up or down. Every y -value increases by k , so each point becomes — the curve rides up the screen, shape intact. the +k lives OUTSIDE the function Shift f(x)=x^2 up 3 : the rule becomes g(x)=x^2+3 . The vertex moves from (0,0) to (0,3) , and every other point rises 3 as well — the point (2,4) lands at (2,7) . 2. Horizontal Shifts: The Sign Flips Now subtract a constant from the input . The graph still slides — but a minus sign moves it right , not left. Watch f(x-3) glide three units to the right : the input has to grow bigger to feed the function the same value it used to get sooner. Graph shifts RIGHT by h — the opposite of the minus sign you see. Graph shifts LEFT by h — a plus inside moves it left. To hit the old output, the new input must be larger to cancel the -h , so the picture slides right. The vertex of (x-3)^2 sits at x=3 , exactly three to the right of the original at x=0 . Why it goes backwards — concretely Let f(x)=x^2 , so (3,9) is on the graph. For g(x)=f(x-2)=(x-2)^2 , where does the output 9 appear? Solve . The value you got at x=3 now shows up at x=5 — two units to the right. The input had to grow to compensate for the subtraction.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.