Translating graphs of functions

 Given y=f(x)

Reflection in the  x- axis  gives y=-f(x)

This has the effect of making the y co-ordinate the negative of whatever it was.

So the point (3, 2) becomes ( 3, -2)

Note that the x co-ordinate is untouched.

 

 

Reflection  in the  y- axis gives y=f(-x)

This has the effect of making the x co-ordinate the negative of whatever it was.

So the point (3, 2) becomes ( -3, 2)

Note that the y co-ordinate is untouched.

 

Translate in the x direction to get y=f(x-c)

This has the effect of shifting the entire graph horizontally.

Remember to go the opposite way to the sign,
so for (x - c) go right c spaces and for (x + c) go left c spaces.

So if f(x) has a point with co-ordinates (3, 2) , then the image of that point on the graph y= f(x+6) is ( -3, 2) and the image on y=f(x-6) is (9,2)

Note that the y co-ordinate is untouched.

 

Translate in the y direction to get y=f(x) + d

This has the effect of shifting the entire graph vertically.

If y=f(x) + d go up d spaces, y=f(x) - d go down d spaces.

So if f(x) has a point with co-ordinates (3, 2) , then the image of that point on the graph y= f(x)+6 is ( 3,8) and the image on y=f(x)-6 is (3, -4)

Note that the x co-ordinate is untouched.

 

• Scale in the x direction to get y=f(x/b) or y=f(bx)

This has the effect of squashing the graph horizontally when b > 1 and stretching it horizontally when b< 1.

If y=f(bx) divide the x cordinate by b. If y=(x/b) multiply the x co-ordinate by b.

So if f(x) has a point with co-ordinates (3, 2) , then the image of that point on the graph y= f(3x) is ( 1,2) and the image on y=f(x/4) is (4, 2)

Note that the y co-ordinate is untouched.

 

Scale in the y direction to get y=af(x) or y=f(x)/a

This has the effect of stretching the graph vertically when a > 1 and squashing it vertically when a < 1.

If y=af(x) multiply the y cordinate by a. If y=(x/a) divide the y co-ordinate by a.

So if f(x) has a point with co-ordinates (3, 2) , then the image of that point on the graph y= 3f(x) is ( 1,6) and the image on y=f(x)/2 is (3, 1)

Note that the x co-ordinate is untouched.

 

 

Putting it all together

so

1. Shift left C units
2. Squeeze horizontally b units
3. Stretch vertically a units
4. translate vertically d units

 

Example

Given y=f(x), draw the graph y = 3f(2x-2) +4

Comparing the graph with the general form

 

starting with y=f(x)

 

Shift right 2 units

Point (-1,0) is translated to (1,0)

Half the x co-ordinate since b=2

Point (1,0) becomes (0.5,0)

so Point (-1,0) is translated to (0.5,0)

Multiply the y co-ordinate by 3, since a = 3

Point (0.5,0) remains (0.5,0)

so Point (-1,0) is still (0.5,0)

Shift the whole graph up 4 spaces, since d = 4

Point (0.5,0) becomes (0.5,4)

so Point (-1,0) is translated to (0.5,4)

Likewise, the point (1,-4)

is translated to ( 3 ,-4)

then (3/2, -4)

then (3/2, -12)

and finally (3/2, -8)

Working backwards example

 

Given the graph of y=2f(4x+2) -6 , find the original co-ordinates of the translated points (-0.5,0) and (-0.25,2)

 

Taking point (-0.5,0)

Taking point (-0.25,2)

 

Giving points (0,3) and (1,4)

 

 

Reflect in the line y=x to get the inverse,

        if it exists

 

Keep it all positive

y=│f(x)│