For the given equation, list the intercepts and test for symmetry. $$ y=x^{2}-x-72 $$

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To analyze the equation $$ y = x^{2} - x - 72 $$, we'll determine its intercepts and examine its symmetry.

### **Intercepts**

1. **Y-Intercept**:
   - **Definition**: The y-intercept occurs where $$ x = 0 $$.
   - **Calculation**:
     $$
     y = (0)^2 - (0) - 72 = -72
     $$
   - **Y-Intercept**: $$ (0, -72) $$

2. **X-Intercepts**:
   - **Definition**: The x-intercepts occur where $$ y = 0 $$.
   - **Equation to Solve**:
     $$
     0 = x^{2} - x - 72
     $$
   - **Solving the Quadratic Equation**:
     $$
     x = \frac{1 \pm \sqrt{1 + 288}}{2} = \frac{1 \pm \sqrt{289}}{2} = \frac{1 \pm 17}{2}
     $$
     - **First Solution**:
       $$
       x = \frac{1 + 17}{2} = \frac{18}{2} = 9
       $$
     - **Second Solution**:
       $$
       x = \frac{1 - 17}{2} = \frac{-16}{2} = -8
       $$
   - **X-Intercepts**: $$ (9, 0) $$ and $$ (-8, 0) $$

### **Symmetry**

1. **Y-Axis Symmetry**:
   - **Test**: Replace $$ x $$ with $$ -x $$ in the equation and see if it remains unchanged.
   - **Check**:
     $$
     y = (-x)^{2} - (-x) - 72 = x^{2} + x - 72 \neq y
     $$
   - **Conclusion**: *Not symmetric* about the y-axis.

2. **Origin Symmetry**:
   - **Test**: Replace $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$.
   - **Check**:
     $$
     -y = (-x)^{2} - (-x) - 72 \implies -y = x^{2} + x - 72 \neq y
     $$
   - **Conclusion**: *Not symmetric* about the origin.

3. **Axis of Symmetry**:
   - **Definition**: For a quadratic equation $$ y = ax^{2} + bx + c $$, the axis of symmetry is the vertical line $$ x = -\frac{b}{2a} $$.
   - **Calculation**:
     $$
     x = -\frac{-1}{2 \times 1} = \frac{1}{2}
     $$
   - **Conclusion**: The graph is symmetric about the **vertical line $$ x = \frac{1}{2} $$**.

### **Summary**

- **Intercepts**:
  - **Y-Intercept**: $$ (0, -72) $$
  - **X-Intercepts**: $$ (9, 0) $$ and $$ (-8, 0) $$
  
- **Symmetry**:
  - The graph is **symmetric** about the **vertical line $$ x = \frac{1}{2} $$**.
  - It is **not symmetric** with respect to the y-axis or the origin.

---

**Graphical Representation:**

Below is a sketch of the quadratic function $$ y = x^{2} - x - 72 $$:

```
^ y
|
|                   *
|                   |
|                   |
|           *       |
|                   |
|                   |
|        (-8,0)     (9,0)
|
+----------------------------> x
          (0,-72)
```

- The vertex lies on the line $$ x = \frac{1}{2} $$.
- The parabola opens upwards since the coefficient of $$ x^{2} $$ is positive.
**Intercepts and Symmetry for $$ y = x^{2} - x - 72 $$:** - **Y-Intercept**: $$ (0, -72) $$ - **X-Intercepts**: $$ (9, 0) $$ and $$ (-8, 0) $$ - **Symmetry**: The graph is symmetric about the vertical line $$ x = \frac{1}{2} $$. It is not symmetric about the y-axis or the origin. --- **Graphical Summary:** - The parabola opens upwards. - It crosses the y-axis at $$ (0, -72) $$. - It crosses the x-axis at $$ (9, 0) $$ and $$ (-8, 0) $$. - The axis of symmetry is the vertical line $$ x = \frac{1}{2} $$.

For the given equation, list the intercepts and test for symmetry. \( y=x^{2}-x-72 \)

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