Certainly! Let's examine the translation of the given equation step by step:
1. Starting Equation:
[tex]\[
y = (x - 5)^2 + 5
\][/tex]
This is a quadratic equation of the form [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] determine the vertex of the parabola.
2. New Equation:
[tex]\[
y = x^2
\][/tex]
This equation can be written as [tex]\( y = (x - 0)^2 + 0 \)[/tex], which is also in the form [tex]\( y = a(x-h)^2 + k \)[/tex].
3. Identify the Vertex of the Original Equation:
The vertex of the original equation [tex]\( y = (x - 5)^2 + 5 \)[/tex] is at [tex]\( (h, k) \)[/tex] or [tex]\( (5, 5) \)[/tex].
4. Identify the Vertex of the New Equation:
The vertex of the new equation [tex]\( y = x^2 \)[/tex] is at [tex]\( (0, 0) \)[/tex].
5. Determine the Horizonal Translation (x-direction):
- In the original equation, the horizontal term inside the parenthesis is [tex]\( x - 5 \)[/tex]. This means the graph is shifted 5 units to the right from the origin (0, 0).
- In the new equation, the horizontal term is [tex]\( x - 0 \)[/tex], indicating no horizontal shift.
Therefore, the change from [tex]\( x - 5 \)[/tex] to [tex]\( x - 0 \)[/tex] shows a translation of 5 units to the right.
6. Determine the Vertical Translation (y-direction):
- In the original equation, the constant term outside the parenthesis is [tex]\( +5 \)[/tex]. This means the graph is shifted 5 units up from the x-axis.
- In the new equation, the constant term is [tex]\( +0 \)[/tex], indicating no vertical shift.
Therefore, the change from [tex]\( +5 \)[/tex] to [tex]\( +0 \)[/tex] shows a translation of 5 units down.
Conclusion:
The graph of the equation [tex]\( y = (x - 5)^2 + 5 \)[/tex] has been translated 5 units to the right and 5 units down to achieve the graph of the equation [tex]\( y = x^2 \)[/tex]. This can be summarized as:
[tex]\[
\text{Translation: } (5, -5)
\][/tex]
