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2.9 D) Vectors
2.9 D) Vectors
We are also able to use column vectors to show the translation of a curve. Below is an example of a column vector.
There are two different elements in the column vector above:
Let’s now describe in words what the column vector at the start is showing. The top element in the column vector shows the horizontal movement (x direction). The top element is 3, which means that the curve is moving 3 to the right. The bottom element shows us the vertical movement (y direction). The bottom element is -2, which means that the curve is moving down by 2. So, overall, this vector moves the curve 3 to the right and 2 down.
- The top element shows the horizontal movement (how much the x value changes by). A positive value for the top element means that the curve is moving towards the right, and a negative value means that the curve is moving towards the left.
- The bottom element shows the vertical movement (how much the y value changes by). A positive value for the bottom element means that the curve moves upwards, and a negative value means that the curve moves downwards.
Let’s now describe in words what the column vector at the start is showing. The top element in the column vector shows the horizontal movement (x direction). The top element is 3, which means that the curve is moving 3 to the right. The bottom element shows us the vertical movement (y direction). The bottom element is -2, which means that the curve is moving down by 2. So, overall, this vector moves the curve 3 to the right and 2 down.
Translations Parallel to the y axis
A translation parallel to the y axis will result in the graph moving up or down. A translation parallel to the y axis will have the following form when f(x) is included.
A translation parallel to the y axis will result in the graph moving up or down. A translation parallel to the y axis will have the following form when f(x) is included.
When the value of a is positive, all of the points on the curve will move up by the value of a. When the value of a is negative, all of the points on the curve will move down by the value of a. The x values remain the same and the y values change by the value of a. We are able to write this translation as a column vector. There is no change in the x values, and this means that the top component in the column vector is 0. The y coordinates change by the value of a and this means that the bottom component of the column vector is the value of a. Therefore, the column vector that describes how the curve f(x) moves to becomes the curve of f(x) + a is:
Let’s have a few examples where we have specific values for a. We do not need to know what the functions are.
Translations Parallel to the x axis
Translations parallel to the x axis result in the curve moving to the left or the right. They occur when something is added or subtracted from the unknown inside the function bracket. For example, suppose that we have the function f(x), a translation parallel to the x axis would have the form:
Translations parallel to the x axis result in the curve moving to the left or the right. They occur when something is added or subtracted from the unknown inside the function bracket. For example, suppose that we have the function f(x), a translation parallel to the x axis would have the form:
A positive value for a results in the curve moving towards the left by the value of a. For example, the translated curve f(x + 2) has a value for a of 2 and it would result in the curve f(x) moving towards the left by 2. A leftwards move by 2 would result in the top component of the column vector being -2, which is negative a (a was 2 and negative a is -2). Therefore, the top component of the column vector for translations that are parallel to the x axis is -a.
The y coordinates for all of the points on the curve remain the same. This means that the bottom component of the column vector is 0.
The column vector for a translation that takes the form f(x + a) is:
The y coordinates for all of the points on the curve remain the same. This means that the bottom component of the column vector is 0.
The column vector for a translation that takes the form f(x + a) is:
Let’s have a few examples.
Note: a negative negative (--) results in a positive; --2 becomes just 2.