The green dotted line in the graph below represents the function f(x)f\left(x\right)f(x). The blue solid line represents the function g(x)g\left(x\right)g(x), which is the function f(x)f\left(x\right)f(x)after it has gone through a shift transformation. Find the equation for g(x)g\left(x\right)g(x).

g(x)=f(x+2)−3g\left(x\right)=f\left(x+2\right)-3g(x)=f(x+2)−3

The green dotted line in the graph below represents the function f ( x ) f\left(x\right) f ( x ) . The blue solid line represents the function g ( x ) g\left(x\right) g ( x ) , which is the function f ( x ) f\left(x\right) f ( x ) after it has gone through a shift transformation. Find the equation for g ( x ) g\left(x\right) g ( x ) .

Alright. So let's give this problem a try. So here we have that the green dotted line in the graph below represents the function f of x, and that's gonna be this curve that we see right here. Now we're told that the blue solid line represents the function g of x, which is the function f of x after it has gone through a shift transformation. We're asked to find the equation for g of x. Now recall that when a function goes through a shift, you start with your original function f of x, and this becomes f of x minus h plus k. This was the general form for a shift that we discussed in our videos on the shift transformation. Now we can see here that our original function started here at the center, 0, and what happened is this graph got shifted to this position. Well, we can see here that it got shifted down 1, 2, 3 units, and we can see here that it got shifted left 1, 2 units. So what we could do here and I said 2 units because it went 2 positions over from the origin. So what we can do is say that the h value which represents the horizontal shift is negative 2, so it shifted left 2 units, and we can say that the k value, which represents the vertical shift, shifted down 3 units, so it's negative 3. So what our function is going to end up looking like is g of x is going to be f of x minus h plus k, where we have f of x minus, and then our h value is negative 2. So we have f of x minus negative 2, plus the k value, which we said here was negative 3. So this is what our shift is going to look like, but we can clean this up algebraically a bit because we can cancel this negative sign right here, these two negative signs. So what this is really going to end up end up being is that g of x is equal to f of x plus 2, and then this is going to be minus 3. So we can just write this as minus 3 since plus negative 3 is the same thing as minus 3. So g of x is going to end up looking like this, and this will be our transform function as an equation. So that is how you can solve this problem. Let's move on.

The green dotted line in the graph below represents the function f(x)f\left(x\right)f(x). The blue solid line represents the function g(x)g\left(x\right)g(x), which is the function f(x)f\left(x\right)f(x) after it has gone through a shift transformation. Find the equation for g(x)g\left(x\right)g(x).

g(x)=f(x+2)−3g\left(x\right)=f\left(x+2\right)-3g(x)=f(x+2)−3

g(x)=f(x−2)+3g\left(x\right)=f\left(x-2\right)+3g(x)=f(x−2)+3

g(x)=f(x−2)−3g\left(x\right)=f\left(x-2\right)-3g(x)=f(x−2)−3

g(x)=f(x)−3g\left(x\right)=f\left(x\right)-3g(x)=f(x)−3

3. Functions

Transformations

College Algebra

3. Functions

Transformations

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Multiple Choice

The green dotted line in the graph below represents the function $f$\left$(x$\right$)$ . The blue solid line represents the function $g$\left$(x$\right$)$ , which is the function $f$\left$(x$\right$)$ after it has gone through a shift transformation. Find the equation for $g$\left$(x$\right$)$ .

$g$\left$(x$\right$)=f$\left$(x-2$\right$)+3$

$g$\left$(x$\right$)=f$\left$(x-2$\right$)-3$

$g$\left$(x$\right$)=f$\left$(x+2$\right$)-3$

$g$\left$(x$\right$)=f$\left$(x$\right$)-3$

Verified step by step guidance

Identify the transformation from the graph: The green dotted line represents f(x), and the blue solid line represents g(x). Notice that the blue line is a horizontal shift to the left and a vertical shift downward from the green line.

Determine the horizontal shift: The graph of g(x) is shifted 2 units to the left compared to f(x). This means the transformation involves replacing x with (x + 2) in the function f(x).

Determine the vertical shift: The graph of g(x) is shifted 3 units downward compared to f(x). This means the transformation involves subtracting 3 from the function f(x).

Combine the transformations: The horizontal shift of 2 units to the left and the vertical shift of 3 units downward can be combined into the transformation g(x) = f(x + 2) - 3.

Verify the transformation: Check that the transformation g(x) = f(x + 2) - 3 correctly describes the shift observed in the graph, ensuring that the blue line is indeed the result of shifting the green line 2 units left and 3 units down.

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