Consider the parabola ( 3) 82y x= + − . Sketch this parabola. What is the equation of theimage if it is:a translated 8 units up and 3 units to the right?

1. Break Down the Problem

1. Identify the given equation of the parabola: $y = \frac{1}{3}(x - 8)^2 - 2$.
2. Determine the changes applied to the parabola: a translation of 8 units up and 3 units to the right.

2. Relevant Concepts

• To translate a parabola:
  • A translation of $h$ units to the right is performed by replacing $x$ with $x - h$.
  • A translation of $k$ units up is performed by adding $k$ to $y$.

3. Analysis and Detail

1. Original Equation: $y = \frac{1}{3}(x - 8)^2 - 2$

2. Translation 3 Units to the Right: Replace $x$ with $x - 3$: $y = \frac{1}{3}((x - 3) - 8)^2 - 2 = \frac{1}{3}(x - 11)^2 - 2$

3. Translation 8 Units Up: Add 8 to the entire equation: $y = \frac{1}{3}(x - 11)^2 - 2 + 8 = \frac{1}{3}(x - 11)^2 + 6$

4. Verify and Summarize

• The new expression for $y$ after translating the parabola is: $y = \frac{1}{3}(x - 11)^2 + 6$

Final Answer

The equation of the translated parabola is: $y = \frac{1}{3}(x - 11)^2 + 6$