Question

question 3 (1 point)
which graph is a possible sketch of the function $f(x) = (x - 6)(x + 3)^3$?
\bigcirc a)
\

$$\begin{tikzpicture}scale=0.5 \\draw-> (-10,0) -- (10,0) noderight {$x$}; \\draw-> (0,-700) -- (0,700) nodeabove {$y$}; \\foreach \\x in {-10,-8,-6,-4,-2,2,4,6,8,10} \\draw (\\x,2pt) -- (\\x,-2pt) nodebelow {$\\x$}; \\foreach \\y in {-625,-500,-375,-250,-125,125,250,375,500,625} \\draw (2pt,\\y) -- (-2pt,\\y) nodeleft {$\\y$}; \\drawthick, smooth, domain=-3.5:9 plot (\\x, { (\\x - 6)*(\\x + 3)^3 }); \\end{tikzpicture}$$

\bigcirc b)
\

$$\begin{tikzpicture}scale=0.5 \\draw-> (-10,0) -- (10,0) noderight {$x$}; \\draw-> (0,-700) -- (0,700) nodeabove {$y$}; \\foreach \\x in {-10,-8,-6,-4,-2,2,4,6,8,10} \\draw (\\x,2pt) -- (\\x,-2pt) nodebelow {$\\x$}; \\foreach \\y in {-625,-500,-375,-250,-125,125,250,375,500,625} \\draw (2pt,\\y) -- (-2pt,\\y) nodeleft {$\\y$}; \\drawthick, smooth, domain=-5:9 plot (\\x, { (\\x - 6)*(\\x + 3)^3 }); \\end{tikzpicture}$$

Explanation:

Step1: Find x-intercepts

Set $f(x)=0$, solve $(x-6)(x+3)^3=0$.
Solutions: $x=6$ (multiplicity 1), $x=-3$ (multiplicity 3)

Step2: Analyze intercept behavior

• For $x=6$ (odd multiplicity 1): graph crosses x-axis.
• For $x=-3$ (odd multiplicity 3): graph crosses x-axis, with a flatter shape near the intercept (due to higher odd multiplicity).

Step3: Test sign of $f(x)$

• For $x < -3$ (e.g., $x=-4$): $f(-4)=(-4-6)(-4+3)^3=(-10)(-1)=10>0$, so graph is above x-axis.
• For $-3 < x < 6$ (e.g., $x=0$): $f(0)=(0-6)(0+3)^3=(-6)(27)=-162<0$, so graph is below x-axis.
• For $x > 6$ (e.g., $x=7$): $f(7)=(7-6)(7+3)^3=(1)(1000)=1000>0$, so graph is above x-axis.

Step4: Match to given graphs

Graph b) is above x-axis when $x < -3$, crosses x-axis at $x=-3$, is below x-axis between $-3$ and $6$, crosses x-axis at $x=6$, and is above x-axis when $x > 6$, which matches all properties.

Answer:

b) <The graph that is above the x-axis for $x < -3$, crosses the x-axis at $x=-3$, dips below the x-axis between $-3$ and $6$, crosses the x-axis at $x=6$, and rises above the x-axis for $x > 6$>