 f (x) = x2 − 6x + 4
(3.1) (2 points) Without completing the square,
find the vertex. 

To find the vertex of a quadratic function \( f(x) = ax^2 + bx + c \) without completing the square, we can use the vertex formula for the x-coordinate:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

For the given function \( f(x) = x^2 - 6x + 4 \), the coefficients are:
- \( a = 1 \)
- \( b = -6 \)

Substituting into the formula:

\[
x_{\text{vertex}} = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3
\]

Now, to find the y-coordinate of the vertex, substitute \( x = 3 \) into the original function \( f(x) \):

\[
f(3) = (3)^2 - 6(3) + 4 = 9 - 18 + 4 = -5
\]

Thus, the vertex of the quadratic function is \( (3, -5) \).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does completing the square confirm the vertex we found?
2. What is the significance of the vertex in the graph of a parabola?
3. Can you explain how the direction of the parabola is determined by the sign of \( a \)?
4. How would you solve for the x-intercepts of this quadratic function?
5. What is the axis of symmetry for this quadratic function?

**Tip:** The vertex represents the maximum or minimum point of a parabola, depending on the direction it opens.

Without completing the square, find the vertex of the quadratic function f(x) = x^2 − 6x + 4.

Discover how to find the vertex of the quadratic function f(x) = x^2 - 6x + 4 using the vertex formula. Learn key algebraic concepts and understand the properties of quadratic functions. Suitable for students in Grades 8-10.

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