The given function is:

$f(x) = x^2 - 8x + 16$

Step 1: Simplifying the Function

This quadratic expression can be simplified further by recognizing it as a perfect square:

$f(x) = (x - 4)^2$

This is because:

$(x - 4)^2 = x^2 - 8x + 16$

Step 2: Analyzing the Function

• Vertex form: The function is now in vertex form, $f(x) = (x - 4)^2$, where the vertex of the parabola is at $(4, 0)$.
• Axis of symmetry: The line of symmetry is $x = 4$.
• Minimum value: Since the square of a number is always non-negative, the minimum value of $f(x)$ occurs at $x = 4$, where $f(x) = 0$.

Step 3: Graphing Insights

• The parabola opens upwards (since the coefficient of $x^2$ is positive).
• The graph touches the x-axis at $x = 4$, indicating that the only root of the equation $f(x) = 0$ is $x = 4$.

Would you like more details on graphing this function or any further analysis?

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Here are 5 related questions:

1. How do you find the vertex of a quadratic function in standard form?
2. What is the importance of recognizing a perfect square trinomial?
3. How does the graph of $f(x) = x^2$ compare to $f(x) = (x-4)^2$?
4. Can you factor non-perfect square quadratic functions similarly?
5. How do transformations affect the graph of a quadratic function?

Tip: The vertex form of a quadratic equation helps you easily identify the vertex and the direction in which the parabola opens.