Question

1. ( f(x)=3(x + 7)^{3}-4 ) and give appropriate windows for each function. provide the information requested. increasing interval: decreasing interval: x - intercept(s): y - intercept(s): end behavior: as ( x\to-infty,f(x)\to ) as ( x\toinfty,f(x)\to ) transformation(s) as compared to ( f(x)=x^{3} )

Explanation:

Step1: Find x - intercept

Set $f(x)=0$, so $3(x + 7)^{3}-4=0$. Then $(x + 7)^{3}=\frac{4}{3}$, and $x=-7+\sqrt[3]{\frac{4}{3}}$.

Step2: Find y - intercept

Set $x = 0$, then $f(0)=3(0 + 7)^{3}-4=3\times343-4=1029 - 4=1025$.

Step3: End - behavior

As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$.

Step4: Increasing/Decreasing

The derivative $f^\prime(x)=9(x + 7)^{2}\geq0$, so the function is increasing on $(-\infty,\infty)$.

Step5: Transformation

Compared to $y = x^{3}$, it is shifted 7 units left, vertically stretched by a factor of 3 and shifted 4 units down.

Answer:

x - intercept: $x=-7+\sqrt[3]{\frac{4}{3}}$; y - intercept: $y = 1025$; As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$; Increasing on $(-\infty,\infty)$; Shifted 7 units left, vertically stretched by 3, 4 units down.