Let’s analyze the problem from the uploaded image. The given quadratic is written in vertex form:

$y = 3(2x - 8)^2 + 5$

We will go step by step to fill out the required information:

1. Vertex:
   The vertex form of a quadratic is: $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex.
   Here, $2x - 8$ can be rewritten as $2(x - 4)$, so $h = 4$ and $k = 5$.
   Vertex: $(4, 5)$.

2. Axis of Symmetry (AOS):
   The axis of symmetry (AOS) is the vertical line that passes through the vertex. It has the equation: $x = h$ So, the AOS is: $x = 4$

3. Range:
   The quadratic opens upwards because the leading coefficient $3 > 0$. The minimum value of $y$ occurs at the vertex, where $y = 5$.
   Thus, the range is: $y \geq 5$

4. Vertical Stretch:
   The coefficient $a = 3$ indicates a vertical stretch by a factor of $3$.

5. Horizontal Compression:
   The term $2x$ indicates a horizontal compression by a factor of the reciprocal of $2$, i.e., $\frac{1}{2}$.

6. Horizontal Shift:
   The term $x - 4$ indicates a horizontal shift to the right by $4$ units.

7. Vertical Shift:
   The $+5$ outside the squared term indicates a vertical shift up by $5$ units.

Final Answers:

1. Vertex: $(4, 5)$
2. AOS: $x = 4$
3. Range: $y \geq 5$
4. Vertical Stretch: $3$
5. Horizontal Compression: $\frac{1}{2}$
6. Horizontal Shift: $4$ units to the right
7. Vertical Shift: $5$ units up

Would you like a detailed breakdown of any step or have additional questions?

5 Related Questions:

1. How do you determine the direction a parabola opens?
2. What does the coefficient inside the parentheses ($2x$) affect in the graph of the parabola?
3. How does a horizontal shift differ from a vertical shift in vertex form?
4. Can the vertex form be converted to standard form? How?
5. What happens if the coefficient $a$ is negative in vertex form?

Tip:

Always rewrite terms like $2x - 8$ into factored form $2(x - 4)$ to clearly identify shifts and transformations.