Horizontal and vertical translations (shifts) of cubic functions are written in the form y=(x-h)3+cy=(x−h)3+c where the point (h,c)(h,c) is the inflection point of the function.
-h−h is a shift to the right and +c+c is a shift upwards.
(-h)→ Shift Right
(+h)← Shift Left
(-c)↓ Shift Down
(+c)↑ Shift Up
To obtain the graph of y=(x-2)3+3, first sketch the function y=x3.
Sketch the function y=x3. Remember the formula y=(x-h)3+c when applied to y=x3 (can be rewritten as y=(x-0)3+0) has its inflection point at (0,0).
Using the formula y=(x-h)3+c for horizontal and vertical translations (shifts) and remembering that the point (h,c) is the inflection point of the function, the inflection point for y=(x-2)3+3 is (2,3).
Sketch the curve for y=(x-2)3+3 through its inflection point (2,3) following the same shape as y=x3.
Horizontal and vertical translations (shifts) of quadratic functions are written in the form y=(x-h)2+c where the point (h,c) is the vertex of the function.
-h is a shift to the right and +c is a shift upwards.
(-h)→ Shift Right
(+h)← Shift Left
(-c)↓ Shift Down
(+c)↑ Shift Up
To obtain the graph of y=x2–6x+11, first sketch the function y=x2.
Sketch the function y=x2. Remember the formula y=(x-h)2+c when applied to y=x2 (can be rewritten as y=(x-0)2+0) has its vertex at (0,0).
Then rewrite y=x2–6x+11 in the standard form by completing the square.
y=(x2-6x+9)-9+11
y=(x-3)2+2.
Remember the standard form is y=(x-h)2+c.
Using the formula y=(x-h)2+c for horizontal and vertical translations (shifts) and remembering that the point (h,c) is the vertex of the function, the vertex point for y=(x-3)2+2 is (3,2).
Sketch the curve for y=(x-3)2+2 through its vertex point (3,2) following the same shape as y=x2.