$3-4$ Match each equation with its graph. Explain your choices.(Don't use a computer or graphing calculator.)$\begin{array}{ll}\text{ (a) }y=3x& \text{ (b) }y=3^x\\ \text{ (c) }y=x^3& \text{ (d) }y=\sqrt[3]{x}\end{array}$

Key Concepts: Graphing Equations, Exponential Functions, Square Root Functions Explanation: To match each equation with its graph, we need to analyze the characteristics and behaviors of each type of functions. Step by Step Solution: Step 1. Let's start with equation (a), which is a linear function with a slope of 3 and a y-intercept of 0. Since the slope is positive and the y-intercept is at the origin, the graph will pass through the first quadrant and increase as we move from left to right. Step 2. Next, we move to equation (b), which is an exponential function with a base of 3. For any exponential function with a base greater than 1, the graph will increase rapidly as we move from left to right and approach the x-axis but never touch or cross it. Therefore, the graph of (b) will increase quickly and stay above the x-axis. Step 3. Then, we consider equation (c), which is a cubic function. It has one point where the slope is zero, at the origin. Then the function increases or decreases monotonically. Therefore, the graph of (c) passes through the origin, and increases or decreases monotonically depending on the sign of the coefficient of the $x^3$ term. Step 4. Finally, we examine equation (d), which is a cube root function. It has its domain to be all real numbers, and its range to be all real numbers. The graph of (d) is continuous and has a cusp at the origin. As we move away from the origin, the graph either increases or decreases depending on whether the coefficient of the cube root is positive or negative respectively. Final Answer: Equation (a) matches with line 4, equation (b) matches with line 3, equation (c) matches with line 2, and equation (d) matches with line 1.