Describe the translation.
y=(x−5)2+5 → y=(x−0)2+0
I think she meant
y=(x−5)^2+5 → y=(x−0)^2+0
shift left 5 and down 5
y = 2 x - 5
to
y = 2 x
moves up 5
To describe the translation from y=(x−5)2+5 to y=(x−0)2+0, we need to look at how the function has changed.
First, let's analyze the original function y=(x−5)2+5:
1. The term (x−5) represents a horizontal shift to the right by 5 units. This means that every point on the graph is shifted 5 units to the right compared to the standard parabola y=x^2.
2. The term (x−5)^2 represents a vertical stretching or compression of the graph. In this case, it does not affect the position of the graph but only alters the shape. It represents a quadratic relationship, resulting in a parabolic shape.
3. The term +5 shifts the entire graph vertically upwards by 5 units. This means that the graph is lifted 5 units higher compared to the standard parabola.
Now let's analyze the updated function y=(x−0)2+0:
1. The term (x−0) does not affect the position of the graph but only alters the shape. It represents a linear relationship, resulting in a straight line.
2. The term (x−0)^2 means that the graph is not modified in terms of shape.
3. The term +0 means that the graph remains at the same vertical level as the standard parabola.
In summary, the translation from y=(x−5)2+5 to y=(x−0)2+0 results in a change from a shifted and lifted parabola to the standard parabola with no horizontal and vertical shifts.
The given expressions are equations of quadratic functions in vertex form.
To understand the translation between the two equations, let's analyze the components:
The original equation is y = (x - 5)^2 + 5.
The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
- In the original equation, the vertex is located at (5, 5), as h = 5 and k = 5.
Now, let's examine the modified equation: y = (x - 0)^2 + 0.
- In this equation, we see that h = 0 and k = 0. Therefore, the vertex is at the origin, (0, 0).
In simpler terms, the translation from the original equation to the modified equation involves shifting the vertex of the parabola from (5, 5) to the origin (0, 0). This means the parabola's vertex was moved 5 units to the left and 5 units up to reach the new position at (0, 0).