Suppose f(x)=x^3 and the graph of g(x) is the graph of f(x) shifted vertically up 1 units and horizontally to the right 3 units.
Find the Formula fir g(x)
f(x)-3 = (x-1)^3
so
f(x) = 3 + (x-1)^3
I mean g(x)
I should have read more caarefully.
Well, if we shift a function vertically up 1 unit, the new function is given by adding 1 to the original function. So, g(x) = f(x) + 1.
And if we shift a function horizontally to the right 3 units, the new function is given by replacing x with (x - 3) in the original function. So, g(x) = f(x - 3) + 1.
Since f(x) = x^3, we can substitute that in:
g(x) = (x - 3)^3 + 1.
So, the formula for g(x) is g(x) = (x - 3)^3 + 1.
Now, that's a juicy equation. It's as if the function g(x) is saying, "I'm just like f(x), but I've shifted 3 units to the right and I'm feeling 1 unit taller." Ah, the wonders of shifting and stretching functions!
To find the formula for the function g(x), which is the graph of f(x) shifted vertically up 1 unit and horizontally to the right 3 units, we need to adjust the formula for f(x) accordingly.
Let's break down the steps one by one:
1. Vertical Shift: To shift a function vertically, we add or subtract a constant to the original function. In this case, g(x) is shifted up 1 unit, so we add 1 to f(x).
2. Horizontal Shift: To shift a function horizontally, we replace x with (x - h), where h represents the amount of the horizontal shift. In this case, g(x) is shifted to the right 3 units, so we replace x with (x - 3).
Combining these steps, the formula for g(x) can be written as:
g(x) = f(x - 3) + 1
Substituting f(x) with its given formula f(x) = x^3, we have:
g(x) = (x - 3)^3 + 1
Therefore, the formula for g(x) is g(x) = (x - 3)^3 + 1.
To find the formula for g(x), which is the result of shifting the graph of f(x) vertically up 1 unit and horizontally to the right 3 units, we need to apply the appropriate transformations to the original function f(x)=x^3.
The vertical shift would be represented by adding a constant value to the original function. In this case, the graph of g(x) is shifted up 1 unit, so we add 1 to f(x). Therefore, the vertical shift is g(x) = f(x) + 1.
The horizontal shift is represented by subtracting the desired amount from x inside the function. In this case, the graph is shifted 3 units to the right, so we replace x in f(x) with (x - 3). Therefore, the horizontal shift is g(x) = f(x - 3) + 1.
Combining the vertical shift and horizontal shift together, the formula for g(x) would be g(x) = f(x - 3) + 1.
So, the formula for g(x) is g(x) = (x - 3)^3 + 1.