B. Given the equation of parabola, find the following:
a. Coordinates of the vertex
b. Equation of axis of symmetry
c. Equation of directrix
d. Coordinates of the focus
e. Opening of the parabola5. (x - 5) = 4y

The given equation of the parabola is:

\[
(x - 5) = 4y
\]

This equation can be rewritten in a more standard form by isolating \( y \). Let’s first transform it and then proceed to find the requested properties.

### Step 1: Rewrite the equation
To make it easier to compare with the standard form of a parabola, let's solve for \( y \):

\[
y = \frac{1}{4}(x - 5)
\]

Now, this is in the form \( y = a(x - h) \), which represents a horizontal parabola. The general form for a horizontal parabola is:

\[
(x - h)^2 = 4p(y - k)
\]

However, in this case, it is a vertically oriented parabola, where \( y \) depends on \( x \).

Now, let’s extract the relevant information step by step.

### Step 2: Vertex
The equation in the form \( y = \frac{1}{4}(x - 5) \) tells us that the vertex is at \( (h, k) = (5, 0) \).

**Answer: The vertex is \( (5, 0) \).**

### Step 3: Axis of symmetry
The axis of symmetry is the vertical line passing through the vertex. Since the vertex is at \( x = 5 \), the axis of symmetry is:

**Answer: \( x = 5 \).**

### Step 4: Equation of the directrix
The formula for the directrix of a parabola is given by \( y = k - \frac{1}{4a} \), where \( a \) is the coefficient of the linear term in the equation \( y = ax + b \).

Here \( a = \frac{1}{4} \), and \( k = 0 \). So, the directrix is:

\[
y = 0 - \frac{1}{4\left(\frac{1}{4}\right)} = -1
\]

**Answer: The equation of the directrix is \( y = -1 \).**

### Step 5: Coordinates of the focus
The focus lies at a distance \( \frac{1}{4a} \) from the vertex, along the axis of symmetry. Since \( a = \frac{1}{4} \), the distance is:

\[
\frac{1}{4\left(\frac{1}{4}\right)} = 1
\]

So, the focus is 1 unit above the vertex at \( (5, 1) \).

**Answer: The coordinates of the focus are \( (5, 1) \).**

### Step 6: Opening of the parabola
Since the equation is of the form \( y = \frac{1}{4}(x - 5) \), and \( a = \frac{1}{4} > 0 \), the parabola opens **upwards**.

**Answer: The parabola opens upwards.**

---

Do you want further details or have any questions? 

Here are some related questions for further practice:
1. What is the general formula for the equation of a parabola opening sideways?
2. How does the value of \( a \) in the equation \( y = ax^2 \) affect the width of the parabola?
3. What is the relationship between the focus and the directrix in a parabola?
4. How would the equation of the parabola change if it opened downwards?
5. What is the geometric significance of the vertex in a parabola?

**Tip:** The sign of the coefficient \( a \) in the parabola's equation determines whether it opens upwards or downwards in the case of vertical parabolas.

Given the equation of a parabola, find the following: a. Coordinates of the vertex, b. Equation of axis of symmetry, c. Equation of directrix, d. Coordinates of the focus, e. Opening of the parabola. The equation provided is (x - 5) = 4y.

Solve the equation of the parabola (x - 5) = 4y by finding the vertex, axis of symmetry, directrix, focus, and opening direction. This problem explores key concepts of conic sections, including the relationship between the focus and directrix. Suitable for students in Grades 9-12.

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