The vertex of a parabola represented by f(x)=x^2-4x+3 has coordinates of (2,-1). Find the coordinates of the vertex of the parabola defined by g(x)=f(x-2). Explain how you arrived to your answer.
My question: Would you move the parabola represented by f(x) 2 spaces to the right to get the parabola represented by g(x)?
yes,
the new vertex will be (4,-1)
proof:
f(x-2) = (x-2)^2 - 4(x-2) + 3
= x^2 - 4x + 4 - 4x + 8 + 3
= x^2 - 8x + 15
= g(x)
vertex of g(x) = -b/(2a)
= 8/2 = 4
g(4) = 4^2 - 8(4) + 15 = -1
vertex is (4,-1)
Well, the parabola represented by f(x)=x^2-4x+3 has its vertex at (2,-1). So, if we want to find the vertex of the parabola represented by g(x)=f(x-2), we need to see how the original parabola f(x) has been shifted.
First, let's analyze what happens when we replace x in f(x) with x - 2. This means that any x-value in the original parabola will be shifted to the right by two units. So, yes, you are correct, we are moving the parabola represented by f(x) two spaces to the right to get the parabola represented by g(x). This is because we are subtracting 2 from x, effectively shifting the graph horizontally.
Since the original vertex (2,-1) has been shifted two units to the right, the new vertex will be at (4,-1) for the parabola represented by g(x)=f(x-2). So, the coordinates of the vertex of g(x) are (4,-1).
In summary, the parabola represented by g(x)=f(x-2) is obtained by shifting the original parabola f(x) two units to the right. Therefore, the vertex of g(x) is found by adding 2 to the x-coordinate of the vertex of f(x), which gives us (4,-1).
To determine the coordinates of the vertex of the parabola defined by g(x) = f(x-2), we need to understand the effect of the transformation on the original vertex (2,-1).
When we substitute x-2 in place of x in the equation f(x), we are essentially shifting the parabola f(x) two units to the right.
Here's how the transformation affects the original vertex (2,-1):
- The x-coordinate of the vertex shifts by 2 units to the right. So, the new x-coordinate becomes 2 + 2 = 4.
- Since the y-coordinate remains unchanged, the new y-coordinate will still be -1.
Therefore, the coordinates of the vertex of the parabola defined by g(x) = f(x-2) are (4, -1).
To summarize, yes, we can say that moving the parabola represented by f(x) two spaces to the right gives us the parabola represented by g(x). This is because replacing x with (x-2) in the equation f(x) causes a horizontal translation of the graph to the right by 2 units.
Yes, you would indeed move the parabola represented by f(x) two spaces to the right to get the parabola represented by g(x). To understand how this works, let's go through the steps to find the coordinates of the vertex of the parabola defined by g(x)=f(x-2).
1. Start with the original equation of f(x) = x^2 - 4x + 3.
2. To transform f(x) into g(x), we need to substitute (x-2) in place of x in the equation. So we have g(x) = f(x-2) = (x-2)^2 - 4(x-2) + 3.
3. Simplify g(x) by expanding and combining like terms. g(x) = x^2 - 4x + 4 - 4x + 8 + 3 = x^2 - 8x + 15.
4. Now we have the equation for g(x). The next step is to find the vertex of g(x) to determine its coordinates.
5. The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). In this case, a = 1, b = -8, and c = 15.
6. To find the x-coordinate of the vertex, substitute the values of a, b, and c into the formula: x = -(-8) / (2*1) = 8/2 = 4.
7. Substitute this value of x into the equation g(x) = x^2 - 8x + 15 to find the y-coordinate of the vertex: g(4) = 4^2 - 8*4 + 15 = 16 - 32 + 15 = -1.
8. Therefore, the coordinates of the vertex of the parabola defined by g(x) = f(x-2) are (4, -1).
In summary, to find the coordinates of the vertex of the parabola defined by g(x)=f(x-2), we substituted (x-2) for x in the original equation of f(x), simplified the equation to g(x), and then used the vertex formula to find the coordinates of the vertex of g(x).