Sketch the gaph \( y=4.3^{3 x} \)

To sketch the graph of the function \( y = 4.3^{3x} \), we can analyze its features step by step: 1. **Identify the type of function**: This is an exponential function of the form \( y = a^{bx} \), where \( a = 4.3 \) and \( b = 3 \). Exponential functions generally have a characteristic shape. 2. **Determine the y-intercept**: The y-intercept occurs when \( x = 0 \): \[ y = 4.3^{3 \cdot 0} = 4.3^0 = 1 \] So, the point \( (0, 1) \) is on the graph. 3. **Behavior as \( x \) approaches negative infinity**: As \( x \) decreases, \( 3x \) becomes very negative, and \( 4.3^{3x} \) approaches 0. Thus, the graph approaches the x-axis but never touches it. 4. **Behavior as \( x \) approaches positive infinity**: As \( x \) increases, \( 3x \) becomes very large, and \( 4.3^{3x} \) increases rapidly towards infinity. 5. **Increasing nature**: Since the base \( 4.3 \) is greater than 1, the function is increasing for all \( x \). 6. **Sketching the graph**: - Start at the point \( (0, 1) \). - As \( x \) moves left, the graph approaches the x-axis (but never touches it). - As \( x \) moves right, the graph rises steeply towards infinity. With these characteristics, you can sketch the graph of \( y = 4.3^{3x} \) as an increasing curve that passes through \( (0, 1) \), approaches the x-axis as \( x \) goes to negative infinity, and rises steeply as \( x \) goes to positive infinity.