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Functions: The function notation is f(x), which means an equation involving x. When y = f(x), it means the same as: for example: y = x
- 2x + 1, or any other equation.
In Graphs(ii) it was seen how equations can be expressed on a graph, and solved graphically. In this section it will be seen what happens to the graph when the whole equation or function is modified by a factor or a term.
When a function is multiplied by a factor the line in the graph will be stretched or squeezed. When a function has an extra term added the line in the graph will be shifted up or down, or left or right.
There are four different examples, each will be shown with the function.
Y Stretch and Squeeze: y = a f(x) Function: y = sin x.
Multiply the function: y = f(x) by 3. New function: y = 3f(x).
Fig 1. y = sin x: y = f(x) y = 3f(x).
The Black line: y = f(x).
The Brown line: y = 3f(x).
The new function stretches the line along the Y axis.
At every point along the X axis the Y value is times 3.
y = a f(x) Function: y = x
Multiply the function: y = f(x) by
. New function: y =
Fig 2. y = x
f(x).
The Black line: y = f(x).
The Brown line: y =
The function squeezes the line along the Y axis.
At every point along the X axis the Y value is halved.
The stretching and squeezing along the Y axis is straightforward; if a function is multiplied by a number greater than 1, the function is stretched; if the number is less than 1, the function is squeezed.
X Stretch and Squeeze: y = f(ax) Function: y = sin x.
Multiply x in the function by 4: New function: y = f(4x).
Fig 3. y = sin x: y = f(x) y = f(4x).
The Black line: y = f(x).
The Brown line: y = f(4x).
The function squeezes the line along the X axis.
There are now 4 times the sin new cycle replacing 1 old cycle.
y = f(ax) Function: y = sin x.
Multiply x in the function by
: New function: y = f(
Fig 4. y = sin x: y = f(x) y = f(
x).
The Black line: y = f(x).
The Brown line: y = f(
The function stretches the line along the X axis.
There is only
The stretching and squeezing along the X axis is not straightforward; if a function is multiplied by a number greater than 1, the function is squeezed; if the number is less than 1, the function is stretched.
Y Shift: y = f(x) + a Function: y = x
Add 4 to the function: New function: y = f(x) + 4.
Fig 5. y = x
The Black line: y = f(x).
The Brown line: y = f(x) + 4.
The function shifts the curve 4 up the Y axis.
The point where the line crosses the Y axis is 3 in the equation y = x
y = f(x) - a Function: y = sin x.
Subtract 3 from the function: New function: y = sin x - 3.
Fig 6. y = sin x y = f(x) y = f(x) - 3.
The Black line: y = f(x).
The Brown line: y = f(x - 3).
The function shifts the cycle 3 down the Y axis.
The point where the line crosses the Y axis was 0, by subtracting 3 from the equation the crossing is now shifted down the Y axis by 3.
If the function has a number to be added or subtracted outside the x brackets it is straightforward, just shift up the Y axis for a plus sign, and shift down the Y axis for a minus sign.
X Shift: y = f(x + a) Function: y = x
- 4x.
Add 5 to x in the function: New function: y = f(x + 5) .
Fig 7. y = x
The Black line: y = f(x).
The Brown line: y = f(x + 5).
The function shifts to the left by 5.
Wherever an x appears in the equation it is replaced by x + 5. Therefore: y = x
y = f(x - a) Function: y = x
Subtract 2 from x in the function: New function: y = f(x - 2) .
Fig 8. y = x
The Black line: y = f(x).
The Brown line: y = f(x - 2).
The function shifts to the right by 2.
If the function has a number to be added or subtracted inside the x brackets it is not straightforward, shift left along the X axis for a plus sign, and shift right along the X axis for a minus sign.
Always remember that an operation done on a function is straightforward for a factor of term outside the (x), it affects the Y axis. An operation with a factor or term inside the (x) brackets are more difficult affecting the X axis, and is opposite in movement to operations on the Y axis.
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