Let's work through the given parabola equation step-by-step:

Given Equation:

$x^2 - 2x + 1 = 8y - 16$

We'll first rewrite it in a more recognizable form for a parabola, and then extract all the necessary properties.

Step 1: Simplify and complete the square

Start by simplifying both sides to make it easier to analyze.

$x^2 - 2x + 1 = 8y - 16$

We can subtract $1$ from both sides:

$x^2 - 2x = 8y - 17$

Next, complete the square on the left-hand side:

$x^2 - 2x = (x - 1)^2$

So, the equation becomes:

$(x - 1)^2 = 8y - 17$

Add 17 to both sides:

$(x - 1)^2 = 8(y - 2.125)$

Now we have the equation in the standard form of a parabola: $(x - h)^2 = 4p(y - k)$ Where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus (and to the directrix).

Step 2: Identify the Vertex

By comparing the equation $(x - 1)^2 = 8(y - 2.125)$ with the standard form, we can see that the vertex is:

Vertex: $(1, 2.125)$

Step 3: Identify the Focus

We know that $4p = 8$, so: $p = \frac{8}{4} = 2$

Since this is a parabola that opens vertically (upward), the focus is 2 units above the vertex.

Focus: $(1, 2.125 + 2) = (1, 4.125)$

Step 4: Find the Directrix

The directrix is a horizontal line $p$ units below the vertex. So, we subtract $2$ from the $y$-coordinate of the vertex.

Directrix: $y = 2.125 - 2 = 0.125$

Step 5: Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. For this equation, it is:

Axis of Symmetry: $x = 1$

Step 6: Endpoints of the Latus Rectum

The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. The length of the latus rectum is $4p = 8$, and since the parabola opens vertically, the endpoints are 4 units horizontally from the focus.

Thus, the endpoints are:

Endpoints of the Latus Rectum: $\left(1 - 4, 4.125\right) = (-3, 4.125)$ $\left(1 + 4, 4.125\right) = (5, 4.125)$

Step 7: Graph

To graph this parabola:

• Plot the vertex at $(1, 2.125)$.
• Plot the focus at $(1, 4.125)$.
• Draw the directrix as the horizontal line $y = 0.125$.
• Draw the axis of symmetry along $x = 1$.
• Mark the endpoints of the latus rectum at $(-3, 4.125)$ and $(5, 4.125)$.
• Sketch the parabola passing through these points, opening upwards.

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Would you like a visual graph representation or any other further details?

Here are some related questions:

1. How do we determine the direction in which a parabola opens?
2. What happens to the shape of the parabola if $p$ is increased or decreased?
3. What is the geometric significance of the latus rectum in parabolas?
4. Can a parabola open horizontally, and how would the equation look in that case?
5. What is the relationship between the vertex, focus, and directrix of a parabola?

Tip: When completing the square, always ensure that you balance both sides of the equation, especially when working with a constant.