The equation $(y-4)^2 = 8(x-3)$ represents a parabola. To understand its properties, we can rewrite it in a more standard form.

Step 1: Identify the Parabola's Standard Form

The standard form of a parabola that opens horizontally is: $(y - k)^2 = 4p(x - h)$ where $(h, k)$ is the vertex of the parabola, and $p$ is the distance from the vertex to the focus (the parabola opens to the right if $p > 0$ or to the left if $p < 0$).

Step 2: Compare with the Given Equation

Given equation: $(y-4)^2 = 8(x-3)$

Here, $k = 4$, $h = 3$, and $4p = 8$. Thus: $p = \frac{8}{4} = 2$

Step 3: Interpret the Results

• Vertex: The vertex of the parabola is at $(3, 4)$.
• Focus: Since $p = 2$, the focus is $2$ units to the right of the vertex. Therefore, the focus is at $(3 + 2, 4) = (5, 4)$.
• Directrix: The directrix is a vertical line $2$ units to the left of the vertex. Thus, the directrix is $x = 3 - 2 = 1$.
• Orientation: Since the equation is in the form $(y-k)^2 = 4p(x-h)$, this parabola opens to the right.

Summary:

• Vertex: $(3, 4)$
• Focus: $(5, 4)$
• Directrix: $x = 1$
• Opening: To the right

Would you like more details or have any questions about this? Here are some related questions you might find helpful:

1. How can I find the length of the latus rectum for this parabola?
2. What is the equation of the parabola if it opens downward instead of to the right?
3. How can I convert this parabola equation into a function of $x$?
4. How would the graph of this equation change if the coefficient 8 was negative?
5. Can you explain the geometric significance of the focus and directrix for this parabola?

Tip: Understanding the standard forms of conic sections helps in quickly identifying the key features like the vertex, focus, and directrix.