15) Tell whether each function is linear. Explain. \( \begin{array}{ll}\text { a) } 2 y=-3 x^{2} & \text { b) } y=4 x-7\end{array} \)

Let's determine whether each given function is linear by analyzing their forms. ### a) \( 2y = -3x^{2} \) **Rearranged Form:** \[ y = \frac{-3}{2}x^{2} \] **Analysis:** - **Degree of the Function:** The highest power of \( x \) is 2, which makes it a **quadratic** function. - **Standard Linear Form:** A linear function has the form \( y = mx + b \), where the highest power of \( x \) is 1. **Conclusion:** This function is **not linear** because the presence of \( x^{2} \) makes it a quadratic function, not a straight line. --- ### b) \( y = 4x - 7 \) **Form:** \[ y = 4x - 7 \] **Analysis:** - **Degree of the Function:** The highest power of \( x \) is 1. - **Standard Linear Form:** It matches the linear form \( y = mx + b \), where: - \( m = 4 \) is the slope. - \( b = -7 \) is the y-intercept. **Conclusion:** This function is **linear** because it fits the standard form of a linear equation, representing a straight line when graphed. --- **Summary:** - **a)** Not linear (it's a quadratic function due to the \( x^{2} \) term). - **b)** Linear (fits the form \( y = mx + b \)).