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

Ask by O'Quinn Alexander. in South Africa
Mar 14,2025

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To sketch the graph of \( y = 4.3^{3x} \), plot the y-intercept at \( (0, 1) \). The graph is an increasing curve that approaches the horizontal asymptote \( y = 0 \) as \( x \) decreases.

Solution

To sketch the graph of the function \( y = 4.3^{3x} \), we can start by understanding the properties of the function. Given function: \( y = 4.3^{3x} \) 1. **Domain**: The domain of the function is all real numbers, since the base of the exponent is a positive number and the exponent is a real number. 2. **Range**: The range of the function is all positive real numbers, since the base of the exponent is greater than 1 and the exponent is a real number. 3. **Asymptotes**: The function has a horizontal asymptote at \( y = 0 \), since as \( x \) approaches negative infinity, \( y \) approaches 0. 4. **Intercepts**: The function has a y-intercept at \( (0, 1) \), since when \( x = 0 \), \( y = 4.3^{3*0} = 1 \). 5. **Increasing/Decreasing**: The function is increasing for all values of \( x \), since the base of the exponent is greater than 1. 6. **Symmetry**: The function is symmetric about the y-axis, since the function is even. 7. **Graph**: The graph of the function will be a curve that starts at the y-intercept and increases as \( x \) increases, approaching the horizontal asymptote at \( y = 0 \) as \( x \) approaches negative infinity. To sketch the graph, you can start by plotting the y-intercept at \( (0, 1) \) and then plot a few more points by substituting different values of \( x \) into the function. You can also use a graphing calculator or software to visualize the graph.

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To sketch the graph of \( y = 4.3^{3x} \), start by noting that this is an exponential function where the base \( 4.3 \) is greater than \( 1 \). This means the function will increase rapidly as \( x \) increases. At \( x = 0 \), \( y = 4.3^{0} = 1 \), which gives you a point at \( (0, 1) \). For negative values of \( x \), the function will approach zero but never actually reach it, creating a horizontal asymptote on the x-axis. This means for \( x < 0 \), the values of \( y \) will be small. For positive \( x \), the curve will rise sharply as the exponent \( 3x \) increases. Plot more points like \( (1, 4.3^3) \) and \( (2, 4.3^6) \) to see how steeply the function climbs. Now, connect these points smoothly, keeping in mind the asymptotic behavior on the left and the rapid growth on the right. Overall, you'll create a curve that starts near the x-axis, passes through \( (0, 1) \), and rises steeply to the right. Happy sketching!

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