To sketch the graph of the function $y = (x - 3)^2 + 5$ without plotting individual points, we can start with the basic graph of the standard quadratic function, $y = x^2$, and apply a series of transformations.

Step-by-Step Process:

1. Start with the graph of $y = x^2$:

   • This is a parabola that opens upwards, with its vertex at the origin (0, 0).

2. Horizontal Shift:

   • The term $(x - 3)^2$ represents a horizontal shift of the graph.
   • Since $x - 3$ means we move the graph 3 units to the right.
   • The new vertex is now at $(3, 0)$.

3. Vertical Shift:

   • The $+5$ outside the squared term represents a vertical shift.
   • This shifts the graph 5 units upwards.
   • The new vertex is now at $(3, 5)$.

4. Final Shape:

   • The graph remains a parabola that opens upwards (since the coefficient of the squared term is positive).
   • The vertex is now located at $(3, 5)$.
   • The shape of the parabola is the same as $y = x^2$, meaning it does not stretch or shrink.

Summary of Transformations:

• Horizontal shift right by 3 units.
• Vertical shift up by 5 units.

With this information, you can sketch the graph of $y = (x - 3)^2 + 5$ as a standard parabola with its vertex at $(3, 5)$.

Do you want any more details on this, or would you like to explore further?

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Relative Questions:

1. How would the graph change if the equation was $y = (x + 3)^2 + 5$?
2. What happens to the graph if we modify it to $y = (x - 3)^2 - 5$?
3. How would a coefficient in front of the squared term (e.g., $y = 2(x - 3)^2 + 5$) affect the graph?
4. Can you describe the graph of $y = -(x - 3)^2 + 5$ and how it compares?
5. What is the axis of symmetry of the graph of $y = (x - 3)^2 + 5$?

Tip:

When dealing with quadratic functions, the vertex form $y = a(x - h)^2 + k$ makes it easy to identify the vertex as $(h, k)$ and apply shifts directly.