How to graph a parabola: Find the vertex, build a function table, and plot points to graph the parabola.

The image illustrates the process of graphing a parabola using the quadratic function \( f(x) = x^2 - 2x - 6 \). Let's break down the steps and explain the solution:

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Step 1: Find the Vertex

The vertex of a parabola given by the quadratic function \( f(x) = ax^2 + bx + c \) can be found using the formula:
\[
x = \frac{-b}{2a}
\]

For the function \( f(x) = x^2 - 2x - 6 \):
- \( a = 1 \)
- \( b = -2 \)
- \( c = -6 \)

Substitute these values into the formula:
\[
x = \frac{-(-2)}{2(1)} = \frac{2}{2} = 1
\]

Now, to find the corresponding \( y \)-coordinate of the vertex, substitute \( x = 1 \) back into the function \( f(x) \):
\[
f(1) = (1)^2 - 2(1) - 6 = 1 - 2 - 6 = -7
\]

Thus, the vertex is:
\[
(1, -7)
\]

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Step 2: Build a Function Table

To plot the parabola accurately, we need additional points. We create a table of values by choosing several \( x \)-values and calculating the corresponding \( y \)-values using the function \( f(x) = x^2 - 2x - 6 \).

Here is the function table provided in the image:

| \( x \) | \( y \) |
|---------|---------|
| \(-1\) | \(-3\) |
| \(0\) | \(-6\) |
| \(1\) | \(-7\) |
| \(2\) | \(-6\) |
| \(3\) | \(-3\) |

Let’s verify one of these points as an example. For \( x = 2 \):
\[
f(2) = (2)^2 - 2(2) - 6 = 4 - 4 - 6 = -6
\]
This matches the table.

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Step 3: Plot Points and Graph

Using the vertex \((1, -7)\) and the points from the function table, we plot these points on a coordinate plane and draw a smooth curve through them. The parabola opens upwards because the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive.

The graph shows:
- The vertex at \((1, -7)\).
- Points \((-1, -3)\), \((0, -6)\), \((2, -6)\), and \((3, -3)\) are plotted.
- A smooth parabolic curve is drawn through these points.

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Final Answer

The process of graphing the parabola \( f(x) = x^2 - 2x - 6 \) involves:
1. Finding the vertex: \((1, -7)\).
2. Building a function table with additional points.
3. Plotting the points and drawing the parabola.

The vertex is clearly marked, and the parabola is symmetric about the vertical line \( x = 1 \).

\[
\boxed{(1, -7)}
\]