practice graphing cube root functions. which is the graph of $f(x)=sqrt3{-x}$?

Answer

Explanation:

Step1: Analyze the function's transformation.

The function is $f(x) = \sqrt[3]{-x}$. This can be seen as a transformation of the parent function $g(x) = \sqrt[3]{x}$. The term $-x$ inside the cube root indicates a reflection of the graph of $g(x)$ across the y-axis. Alternatively, $f(x) = \sqrt[3]{-x} = -\sqrt[3]{x}$, which is a reflection of $g(x)$ across the x-axis. For the cube root function, these two reflections result in the same graph. The graph of $y=\sqrt[3]{x}$ is an increasing function passing through the origin. Thus, $f(x) = \sqrt[3]{-x}$ will be a decreasing function passing through the origin.

Step2: Determine coordinates of key points.

To confirm the transformation, we can evaluate the function $f(x) = \sqrt[3]{-x}$ for several values of $x$: If $x = 0$, $f(0) = \sqrt[3]{-0} = 0$. So, the point $(0,0)$ is on the graph. If $x = 1$, $f(1) = \sqrt[3]{-1} = -1$. So, the point $(1,-1)$ is on the graph. If $x = 8$, $f(8) = \sqrt[3]{-8} = -2$. So, the point $(8,-2)$ is on the graph. If $x = -1$, $f(-1) = \sqrt[3]{-(-1)} = \sqrt[3]{1} = 1$. So, the point $(-1,1)$ is on the graph. If $x = -8$, $f(-8) = \sqrt[3]{-(-8)} = \sqrt[3]{8} = 2$. So, the point $(-8,2)$ is on the graph.

Step3: Identify the correct graph using these points.

We compare these key points with the provided graphs (let's label them A, B, C, D from left to right): Graph A: Passes through $(0,0)$, $(8,2)$, and $(-8,-2)$. This represents $y=\sqrt[3]{x}$. Incorrect. Graph B: Passes through $(0,0)$, $(1,-1)$, $(8,-2)$, $(-1,1)$, and $(-8,2)$. This matches all calculated points. Correct. Graph C: Passes through $(0,0)$, $(8,2)$, and $(-8,2)$. This represents $y=\sqrt[3]{|x|}$. Incorrect. Graph D: Passes through $(0,0)$, $(8,-2)$, and $(-8,-2)$. This represents $y=-\sqrt[3]{|x|}$. Incorrect. Therefore, the second graph from the left is the correct representation of $f(x) = \sqrt[3]{-x}$.

Answer:

B. The second graph from the left, which is a decreasing function passing through $(0,0)$, $(1,-1)$, $(8,-2)$, $(-1,1)$, and $(-8,2)$.