Mastering Vertical Translations: How to Shift Graphs Down in Mathematics

vertical translation down

In mathematics, a vertical translation refers to shifting a graph or an object vertically up or down

In mathematics, a vertical translation refers to shifting a graph or an object vertically up or down. The term “down” suggests a negative translation, meaning that the object is shifted downwards. Let’s explore how this concept works.

Consider a basic function, f(x), which represents a graph on a coordinate plane. To move this graph vertically down, we need to subtract a constant value from the function. The new function, denoted as g(x), can be written as g(x) = f(x) – c, where c is the constant by which we want to translate the graph downwards.

To understand this concept better, let’s look at an example. Consider the function f(x) = x^2, which is a simple parabolic curve. If we want to shift this graph two units downwards, we can use the formula g(x) = x^2 – 2.

Let’s see how this affects the graph. If we evaluate f(x) and g(x) for various x-values, we can compare the outputs to observe the vertical translation:

For f(x):
– When x = -1, f(-1) = (-1)^2 = 1
– When x = 0, f(0) = (0)^2 = 0
– When x = 1, f(1) = (1)^2 = 1

For g(x):
– When x = -1, g(-1) = (-1)^2 – 2 = 1 – 2 = -1
– When x = 0, g(0) = (0)^2 – 2 = 0 – 2 = -2
– When x = 1, g(1) = (1)^2 – 2 = 1 – 2 = -1

As you can see, the values of g(x) are two units less than the corresponding values of f(x). This downward shift is reflected in the graph. Every point on the original parabola f(x) has been moved downwards by two units to create the new parabola g(x).

It’s essential to note that if the constant c is positive, it would result in moving the graph upwards rather than downwards. Conversely, if c is negative, as in our example, it leads to a downward shift.

Remember that a vertical translation can be applied to any type of function, not just parabolas. The same principle of adding or subtracting a constant from the function applies to other types of graphs as well.

I hope this explanation helps you understand vertical translations and how they relate to moving graphs downward. Let me know if there’s anything else I can assist you with!

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