To graph the parabola and determine the required characteristics, we first need to analyze the given information:

1. Vertex: $(-3, 1)$
2. Focus: $(-7, 1)$

Step 1: Determine the orientation of the parabola

Since the vertex and focus have the same $y$-coordinate but different $x$-coordinates, the parabola opens horizontally. Because the focus is to the left of the vertex, the parabola opens to the left.

Step 2: Write the equation of the parabola

The general form of a horizontally oriented parabola is:

$(y - k)^2 = 4p(x - h)$

where $(h, k)$ is the vertex, and $p$ is the distance from the vertex to the focus.

Given:

• Vertex: $(h, k) = (-3, 1)$
• Focus: $(-7, 1)$

To find $p$: $p = x_{\text{focus}} - x_{\text{vertex}} = -7 - (-3) = -4$

Substituting these into the equation: $(y - 1)^2 = 4(-4)(x + 3)$

Simplified: $(y - 1)^2 = -16(x + 3)$

Step 3: Find the Directrix

The directrix is a vertical line located at a distance $p$ to the right of the vertex (since the parabola opens to the left):

$x = h + p = -3 + 4 = 1$

So, the equation of the directrix is: $x = 1$

Step 4: Length of the Latus Rectum (LR)

The length of the latus rectum is given by $|4p|$:

$\text{Length of LR} = |4p| = |4(-4)| = 16$

Step 5: Endpoints of the Latus Rectum

The latus rectum is horizontal and passes through the focus $(-7, 1)$. Since the length is 16, the endpoints are 8 units above and below the focus:

$\text{Endpoints} = (-7, 1 + 8) \text{ and } (-7, 1 - 8)$ $\text{Endpoints} = (-7, 9) \text{ and } (-7, -7)$

Step 6: Axis of Symmetry

The axis of symmetry is a horizontal line passing through the vertex:

$y = 1$

Summary

• Equation of the parabola: $(y - 1)^2 = -16(x + 3)$
• Directrix: $x = 1$
• Length of Latus Rectum: 16
• Endpoints of Latus Rectum: $(-7, 9)$ and $(-7, -7)$
• Axis of Symmetry: $y = 1$

Would you like a detailed explanation of any step or the graph of this parabola?

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Follow-up Questions:

1. What are the general steps to derive the equation of a parabola given its vertex and focus?
2. How does the orientation of a parabola affect its equation?
3. Can you find the vertex form of a parabola given only its standard form?
4. How does the distance between the focus and vertex influence the shape of the parabola?
5. What is the significance of the directrix in relation to the parabola?
6. Can a parabola have a vertical directrix? If so, under what conditions?
7. How would the graph of the parabola change if the focus were at $(-3, 5)$?
8. What is the relationship between the latus rectum and the focus of a parabola?

Tip: Always check the orientation of the parabola first by comparing the coordinates of the vertex and focus. This helps in writing the correct equation.