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To solve this problem, we need to:

1. Identify the **vertex** of the parabola.
2. Plot two points on either side of the vertex.

### Step 1: Equation of the Parabola
The given equation is:
\[
y = x^2 - 2x - 1
\]
This is a quadratic equation, and the graph of this equation is a parabola. 

### Step 2: Vertex of the Parabola
To find the vertex, we use the vertex formula for a quadratic equation \( y = ax^2 + bx + c \), where:
- \( a = 1 \),
- \( b = -2 \),
- \( c = -1 \).

The x-coordinate of the vertex is given by:
\[
x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1
\]

Now substitute \( x = 1 \) into the equation to find the y-coordinate:
\[
y = (1)^2 - 2(1) - 1 = 1 - 2 - 1 = -2
\]
So, the **vertex** is at \( (1, -2) \).

### Step 3: Choosing Points on Either Side
We will now choose two points to the left of \( x = 1 \) (for example \( x = 0 \) and \( x = -1 \)) and two points to the right (for example \( x = 2 \) and \( x = 3 \)).

#### For \( x = 0 \):
\[
y = (0)^2 - 2(0) - 1 = -1
\]
Point: \( (0, -1) \)

#### For \( x = -1 \):
\[
y = (-1)^2 - 2(-1) - 1 = 1 + 2 - 1 = 2
\]
Point: \( (-1, 2) \)

#### For \( x = 2 \):
\[
y = (2)^2 - 2(2) - 1 = 4 - 4 - 1 = -1
\]
Point: \( (2, -1) \)

#### For \( x = 3 \):
\[
y = (3)^2 - 2(3) - 1 = 9 - 6 - 1 = 2
\]
Point: \( (3, 2) \)

### Step 4: Plotting the Points
You can now plot the five points:
1. \( (1, -2) \) (the vertex),
2. \( (0, -1) \),
3. \( (-1, 2) \),
4. \( (2, -1) \),
5. \( (3, 2) \).

Would you like me to provide more details on graphing this or discuss any part of the solution?

Here are 5 related questions:
1. What is the axis of symmetry for this parabola?
2. How do you determine if a parabola opens upward or downward?
3. What is the role of the coefficient \( a \) in a quadratic equation?
4. Can you determine the y-intercept directly from the equation?
5. How would the graph change if \( b = 0 \) in the equation?

**Tip:** The vertex is always the minimum or maximum point on the parabola, depending on whether it opens upward or downward.

Plot five points on the parabola: y = x^2 - 2x - 1. The points include the vertex, two points to the left, and two points to the right.

Learn how to plot the vertex and four additional points on the parabola of y = x^2 - 2x - 1. This example covers key algebraic concepts such as quadratic equations and the vertex theorem. Suitable for students in Grades 8-10.

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