(a) Which is the graph of $$ f(x)=-(x+3)^{2}(x-1) \ ? $$

Question

UpStudy AI · Accepted Answer

1. The function is 
   $$
   f(x)=-(x+3)^2 (x-1).
   $$

2. The zeros are found by setting 
   $$
   f(x)=0.
   $$
   This gives:
   - $$x+3=0 \Rightarrow x=-3$$ (with multiplicity 2, so the graph touches and turns at $$-3$$);
   - $$x-1=0 \Rightarrow x=1$$ (the graph crosses the axis at $$1$$).

3. The degree of $$ f(x) $$ is 3 (a cubic). To see the leading behavior, note that when expanded the leading term is 
   $$
   -x^3,
   $$
   so the leading coefficient is negative.

4. The end behavior follows from the leading term:
   - As $$x\to \infty$$, $$f(x)\to -\infty$$;
   - As $$x\to -\infty$$, $$f(x)\to \infty$$.

5. To get an additional point, evaluate $$ f(0) $$:
   $$
   f(0) = -((0+3)^2(0-1)) = -\bigl(9 \cdot (-1)\bigr)=9.
   $$
   So the graph passes through $$(0,9)$$.

6. Summarizing the characteristics:
   - The graph is a cubic curve that crosses the x-axis at $$x=1$$ and touches it at $$x=-3$$ (due to a double zero).
   - As $$x\to -\infty$$, $$f(x)$$ goes to $$\infty$$, and as $$x\to \infty$$, $$f(x)$$ goes to $$-\infty$$.
   - The graph passes through the point $$(0,9)$$.

Thus, the correct graph of $$f(x)=-(x+3)^2 (x-1)$$ is the one that shows these characteristics.
The graph of $$ f(x)=-(x+3)^2 (x-1) $$ is a cubic curve that touches the x-axis at $$ x=-3 $$ and crosses it at $$ x=1 $$. It passes through the point $$(0,9)$$, and as $$ x $$ approaches negative infinity, $$ f(x) $$ goes to positive infinity, while as $$ x $$ approaches positive infinity, $$ f(x) $$ goes to negative infinity.

(a) Which is the graph of \( f(x)=-(x+3)^{2}(x-1) \ ? \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Pre Calculus Questions