Determine whether the equation is linear or not. Then graph the equation by finding and plotting ordered-pair solutions. \( y=x^{2}-6 \)

To determine whether the equation \( y = x^2 - 6 \) is linear or not, we need to analyze its form. A linear equation can be expressed in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the given equation, \( y = x^2 - 6 \), the term \( x^2 \) indicates that this is a quadratic equation, not a linear one. Therefore, the equation is **not linear**. ### Graphing the Equation To graph the equation, we can find some ordered-pair solutions by substituting different values for \( x \) and calculating the corresponding \( y \) values. Let's choose a few values for \( x \): 1. **When \( x = -3 \)**: \[ y = (-3)^2 - 6 = 9 - 6 = 3 \quad \Rightarrow \quad (-3, 3) \] 2. **When \( x = -2 \)**: \[ y = (-2)^2 - 6 = 4 - 6 = -2 \quad \Rightarrow \quad (-2, -2) \] 3. **When \( x = -1 \)**: \[ y = (-1)^2 - 6 = 1 - 6 = -5 \quad \Rightarrow \quad (-1, -5) \] 4. **When \( x = 0 \)**: \[ y = 0^2 - 6 = 0 - 6 = -6 \quad \Rightarrow \quad (0, -6) \] 5. **When \( x = 1 \)**: \[ y = 1^2 - 6 = 1 - 6 = -5 \quad \Rightarrow \quad (1, -5) \] 6. **When \( x = 2 \)**: \[ y = 2^2 - 6 = 4 - 6 = -2 \quad \Rightarrow \quad (2, -2) \] 7. **When \( x = 3 \)**: \[ y = 3^2 - 6 = 9 - 6 = 3 \quad \Rightarrow \quad (3, 3) \] ### Ordered-Pair Solutions The ordered pairs we found are: - \( (-3, 3) \) - \( (-2, -2) \) - \( (-1, -5) \) - \( (0, -6) \) - \( (1, -5) \) - \( (2, -2) \) - \( (3, 3) \) ### Plotting the Points You can plot these points on a coordinate plane. The graph of the equation \( y = x^2 - 6 \) will be a parabola that opens upwards, with its vertex at the point \( (0, -6) \). ### Summary - The equation \( y = x^2 - 6 \) is **not linear**; it is a quadratic equation. - The graph is a parabola, and the ordered pairs provide points to plot on the graph.