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Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. \( f(x)=-(x+1)^{3}-4 \) Use the graphing tool to graph the equation. The domain of \( f(x) \) is (Type your answer in interval notation.)

Ask by Boyd Sherman. in the United States
Feb 26,2025

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The function \( f(x) = -(x + 1)^3 - 4 \) is a cubic function shifted left by 1 unit, reflected across the x-axis, and shifted downward by 4 units. Its domain is all real numbers, written in interval notation as \( (-\infty, \infty) \).

Solución

Step 1. Begin with the basic function y = x³. Step 2. Shift left by 1: Replace x with (x + 1) to get y = (x + 1)³. Step 3. Reflect across the x-axis: Multiply by –1 to obtain y = -(x + 1)³. Step 4. Shift downward by 4: Subtract 4 to arrive at y = -(x + 1)³ – 4. Graphing these transformations shows a cubic curve that has been shifted and flipped compared to the standard cubic. Domain: Since cubic functions are defined for all real numbers and these transformations do not restrict the x-values, the domain is (–∞, ∞).

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Bonus Knowledge

To graph the function \( f(x) = -(x+1)^3 - 4 \), start with the basic cubic function \( g(x) = x^3 \). First, shift the graph left by 1 unit to account for the \( (x+1) \) term. Next, reflect the graph across the x-axis due to the negative sign in front, resulting in \( h(x) = -(x+1)^3 \). Lastly, shift the entire graph down by 4 units for the final transformation. The domain of \( f(x) \) is all real numbers, which in interval notation is \( (-\infty, \infty) \). The range, considering the transformations, is all real numbers less than or equal to -4, written in interval notation as \( (-\infty, -4] \).

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