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

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.