Determine whether the equation is linear or not. Then graph the equation by finding and plotting ordered-pair solutions. \( y=7 x^{3} \)

To determine whether the equation \( y = 7x^3 \) is linear, 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 equation \( y = 7x^3 \), the term \( x^3 \) indicates that this is a cubic function, not a linear function. Therefore, the equation is **not linear**. ### Graphing the Equation To graph the equation \( y = 7x^3 \), we can find some ordered-pair solutions by substituting different values of \( x \) and calculating the corresponding \( y \) values. Let's choose a few values for \( x \): 1. **When \( x = -2 \)**: \[ y = 7(-2)^3 = 7(-8) = -56 \quad \Rightarrow \quad (-2, -56) \] 2. **When \( x = -1 \)**: \[ y = 7(-1)^3 = 7(-1) = -7 \quad \Rightarrow \quad (-1, -7) \] 3. **When \( x = 0 \)**: \[ y = 7(0)^3 = 0 \quad \Rightarrow \quad (0, 0) \] 4. **When \( x = 1 \)**: \[ y = 7(1)^3 = 7(1) = 7 \quad \Rightarrow \quad (1, 7) \] 5. **When \( x = 2 \)**: \[ y = 7(2)^3 = 7(8) = 56 \quad \Rightarrow \quad (2, 56) \] Now we have the following ordered pairs to plot: - \( (-2, -56) \) - \( (-1, -7) \) - \( (0, 0) \) - \( (1, 7) \) - \( (2, 56) \) ### Plotting the Points You can plot these points on a coordinate plane: - The point \( (-2, -56) \) will be very low on the graph. - The point \( (-1, -7) \) will also be below the x-axis. - The point \( (0, 0) \) is the origin. - The point \( (1, 7) \) is above the x-axis. - The point \( (2, 56) \) will be very high on the graph. ### Shape of the Graph The graph of \( y = 7x^3 \) will show a characteristic cubic curve, which passes through the origin and rises steeply in the first quadrant while falling steeply in the third quadrant. The curve will be symmetric about the origin, indicating that it is an odd function. ### Conclusion The equation \( y = 7x^3 \) is not linear, and its graph is a cubic curve.