graph the circles whose equation are given . label each circles's centre and intercepts (if any) with their coordinates pairs   x^2 + y^2 -4x - (9/4) = 0

We are given the equation of a circle:

\[
x^2 + y^2 - 4x - \frac{9}{4} = 0
\]

### Step 1: Rearrange the equation into the standard form of a circle
The general form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where \((h, k)\) is the center of the circle, and \(r\) is the radius. To transform the given equation into this standard form, we will complete the square for the \(x\)-terms.

#### Step 2: Completing the square for the \(x\)-terms
We begin by isolating the \(x\)- and \(y\)-terms:

\[
x^2 - 4x + y^2 = \frac{9}{4}
\]

Now, complete the square for the \(x\)-terms. Take half of the coefficient of \(x\), square it, and add it to both sides of the equation.

\[
x^2 - 4x \quad \text{Half of \(-4\) is \(-2\), and \((-2)^2 = 4\)}
\]

So, we add 4 to both sides:

\[
(x^2 - 4x + 4) + y^2 = \frac{9}{4} + 4
\]

The left side becomes:

\[
(x - 2)^2 + y^2 = \frac{9}{4} + \frac{16}{4} = \frac{25}{4}
\]

Now, we have the equation in standard form:

\[
(x - 2)^2 + y^2 = \left(\frac{5}{2}\right)^2
\]

### Step 3: Identify the center and radius
From the equation \((x - 2)^2 + y^2 = \left(\frac{5}{2}\right)^2\), we can identify:
- The center of the circle is at \((2, 0)\).
- The radius of the circle is \(\frac{5}{2}\) or 2.5 units.

### Step 4: Find the intercepts
#### x-intercepts:
The x-intercepts occur when \(y = 0\). Substitute \(y = 0\) into the equation:

\[
(x - 2)^2 = \left(\frac{5}{2}\right)^2
\]

Taking the square root of both sides:

\[
x - 2 = \pm \frac{5}{2}
\]

So,

\[
x = 2 \pm \frac{5}{2}
\]

This gives two x-intercepts:

\[
x = 2 + \frac{5}{2} = \frac{9}{2}, \quad x = 2 - \frac{5}{2} = -\frac{1}{2}
\]

Thus, the x-intercepts are \(\left(\frac{9}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\).

#### y-intercepts:
The y-intercepts occur when \(x = 0\). Substitute \(x = 0\) into the equation:

\[
(0 - 2)^2 + y^2 = \left(\frac{5}{2}\right)^2
\]

\[
4 + y^2 = \frac{25}{4}
\]

Solving for \(y^2\):

\[
y^2 = \frac{25}{4} - 4 = \frac{25}{4} - \frac{16}{4} = \frac{9}{4}
\]

Taking the square root of both sides:

\[
y = \pm \frac{3}{2}
\]

Thus, the y-intercepts are \((0, \frac{3}{2})\) and \((0, -\frac{3}{2})\).

### Step 5: Graphing the Circle
- The center is \((2, 0)\).
- The radius is \(2.5\).
- The x-intercepts are \(\left(\frac{9}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\).
- The y-intercepts are \((0, \frac{3}{2})\) and \((0, -\frac{3}{2})\).

Would you like a graph of this circle, or do you want further clarifications?

---

**Related Questions:**
1. How do you complete the square for any general quadratic equation?
2. What is the general formula to find intercepts for a circle?
3. How would the equation change if the circle was centered at a different point?
4. How do you determine the points where a circle intersects the axes?
5. How can we find tangent lines to a circle at given points?

**Tip:** Completing the square is a useful technique for transforming quadratic equations into standard forms, often revealing geometric properties directly.

Learn how to graph the circle defined by the equation x^2 + y^2 - 4x - 9/4 = 0. This step-by-step solution covers completing the square, identifying the circle's center and radius, and finding its x- and y-intercepts. Ideal for students in Grades 9-11 studying geometry and circle equations.

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