Step 1: Complete the Square

We start by expressing the quadratic function in vertex form by completing the square.

Given: $f(x) = x^2 - 6x + 8$

1. Isolate the quadratic and linear terms: $f(x) = (x^2 - 6x) + 8$

2. To complete the square, take half of the coefficient of $x$, square it, and add it inside the parentheses. The coefficient of $x$ is $-6$, so: $\frac{-6}{2} = -3, \quad (-3)^2 = 9$

Add and subtract $9$ inside the parentheses to balance the equation: $f(x) = (x^2 - 6x + 9 - 9) + 8$

3. Factor the perfect square trinomial: $f(x) = (x - 3)^2 - 9 + 8$ $f(x) = (x - 3)^2 - 1$

Now the quadratic is in vertex form: $f(x) = (x - 3)^2 - 1$

Step 2: Graph the Function Using Shifting

The vertex form of the quadratic function is: $f(x) = (x - 3)^2 - 1$ This tells us that the graph is a parabola with its vertex at $(3, -1)$ and opens upwards (since the coefficient of $(x - 3)^2$ is positive).

• Shift right by 3 units (from the standard vertex at $(0, 0)$).
• Shift down by 1 unit.

Summary:

• The vertex is at $(3, -1)$.
• The parabola opens upwards, and the axis of symmetry is $x = 3$.

Would you like to see more details about graphing or have any questions?

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Related Questions:

1. How do you determine the vertex from the completed square form?
2. What is the axis of symmetry for the function?
3. How does the direction of the parabola's opening change based on the coefficient?
4. How does the transformation affect the graph of a quadratic function?
5. What is the minimum value of $f(x)$ and where does it occur?

Tip: Always check if the quadratic is factorable before completing the square, as it can save time in solving!