ACTIVITY #4 PARABOLA

This activity will be our last official activity for the 1 quarter. This will be answered on a piece of graphing paper. This can be answered by solo or with a chosen pair/trio and don't forget to write your name/the names of your partner. This activity will have a total of 20 points. Goodluck!

On the given parabola equations, find the following:

a) Vertex

b) Focus

c) Endpoints of Latus Rectum

d) Directrix

e) Axis of Symmetry

g) Graph

1. x²-2x+1=8y-16


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.

---

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.

ACTIVITY #4 PARABOLA

This activity will be our last official activity for the 1 quarter. This will be answered on a piece of graphing paper. This can be answered by solo or with a chosen pair/trio and don't forget to write your name/the names of your partner. This activity will have a total of 20 points. Goodluck!

On the given parabola equations, find the following:

a) Vertex

b) Focus

c) Endpoints of Latus Rectum

d) Directrix

e) Axis of Symmetry

g) Graph

1. x²-2x+1=8y-16

Solve the parabola equation x²-2x+1=8y-16 by finding the vertex, focus, directrix, axis of symmetry, and endpoints of the latus rectum. This detailed step-by-step solution covers the key concepts of graphing parabolas and understanding their properties. Suitable for high school students in Grades 10-12.

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