Answer :
We wish to understand and describe the graph of the cubic function
[tex]$$
f(x)=x^3+9x^2+2x-48.
$$[/tex]
Below is a detailed step-by-step explanation.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 1. Factorizing the Polynomial
One very useful step in graphing a polynomial is to factor it in order to identify its roots. In this case, the polynomial factors as
[tex]$$
f(x) = (x-2)(x+3)(x+8).
$$[/tex]
This factorization immediately tells us that the function has three real roots (or zeros) given by:
[tex]$$
x=2,\quad x=-3,\quad \text{and} \quad x=-8.
$$[/tex]
These are the points where the graph crosses the [tex]$x$[/tex]‑axis.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 2. Analyzing the End Behavior
Since [tex]$f(x)$[/tex] is a cubic polynomial with a positive leading coefficient, its end behavior is as follows:
- As [tex]$x\to \infty$[/tex], [tex]$f(x)\to \infty$[/tex].
- As [tex]$x\to -\infty$[/tex], [tex]$f(x)\to -\infty$[/tex].
This means the graph falls to the left and rises to the right.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 3. Finding the Critical Points
To determine the locations of local maximum and minimum values (i.e. the turning points of the graph), we compute the derivative of [tex]$f(x)$[/tex]:
[tex]$$
f'(x)=\frac{d}{dx}\left(x^3+9x^2+2x-48\right)=3x^2+18x+2.
$$[/tex]
We then solve the equation
[tex]$$
3x^2+18x+2=0
$$[/tex]
for [tex]$x$[/tex]. The exact solutions are:
[tex]$$
x=-3-\frac{5\sqrt{3}}{3}\quad \text{and}\quad x=-3+\frac{5\sqrt{3}}{3}.
$$[/tex]
For clarity, their numerical approximations are approximately:
[tex]$$
x\approx -5.887\quad \text{and}\quad x\approx -0.113.
$$[/tex]
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 4. Classifying the Critical Points
To decide whether these critical points correspond to local maxima or minima, we examine the second derivative:
[tex]$$
f''(x)=\frac{d}{dx}(3x^2+18x+2)=6x+18.
$$[/tex]
- At [tex]$x\approx -5.887$[/tex], we have
[tex]$$
f''(-5.887) \approx 6(-5.887)+18\approx -35.322+18\approx -17.322.
$$[/tex]
Since [tex]$f''(-5.887)<0$[/tex], this critical point is a local maximum.
- At [tex]$x\approx -0.113$[/tex], we have
[tex]$$
f''(-0.113) \approx 6(-0.113)+18\approx -0.678+18\approx 17.322.
$$[/tex]
Since [tex]$f''(-0.113)>0$[/tex], this critical point is a local minimum.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 5. Summary of the Graph’s Features
Based on the analysis, the graph of
[tex]$$
f(x)=x^3+9x^2+2x-48
$$[/tex]
has the following characteristics:
1. Zeros of the Function:
The graph crosses the [tex]$x$[/tex]-axis at
[tex]$$
x=-8,\quad x=-3,\quad \text{and} \quad x=2.
$$[/tex]
2. End Behavior:
As [tex]$x\to -\infty$[/tex], [tex]$f(x)\to -\infty$[/tex], and as [tex]$x\to \infty$[/tex], [tex]$f(x)\to \infty$[/tex].
3. Critical Points:
- A local maximum at [tex]$x\approx -5.887$[/tex], and
- A local minimum at [tex]$x\approx -0.113$[/tex].
4. Y-Intercept:
When [tex]$x=0$[/tex],
[tex]$$
f(0)=0^3+9\cdot0^2+2\cdot0-48=-48.
$$[/tex]
So the graph crosses the [tex]$y$[/tex]-axis at [tex]$(0,-48)$[/tex].
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 6. Describing the Graph
When you graph the function on a tool such as Desmos, you will observe:
- The curve crosses the [tex]$x$[/tex]‑axis at [tex]$x=-8$[/tex], [tex]$x=-3$[/tex], and [tex]$x=2$[/tex].
- You will notice a turning point (local maximum) near [tex]$x\approx -5.887$[/tex] and a turning point (local minimum) near [tex]$x\approx -0.113$[/tex].
- The overall shape is typical of a cubic polynomial with the left end descending to [tex]$-\infty$[/tex] and the right end rising to [tex]$\infty$[/tex].
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Conclusion
The graph of
[tex]$$
f(x)=x^3+9x^2+2x-48
$$[/tex]
can be completely described by its factored form
[tex]$$
(x-2)(x+3)(x+8),
$$[/tex]
its zeros at [tex]$x=-8$[/tex], [tex]$-3$[/tex], and [tex]$2$[/tex], the end behavior as [tex]$x\to\pm\infty$[/tex], and its critical points located approximately at [tex]$x\approx -5.887$[/tex] (local maximum) and [tex]$x\approx -0.113$[/tex] (local minimum). Plotting the function on a graphing utility will confirm these features and display the characteristic cubic “S”-shape.
[tex]$$
f(x)=x^3+9x^2+2x-48.
$$[/tex]
Below is a detailed step-by-step explanation.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 1. Factorizing the Polynomial
One very useful step in graphing a polynomial is to factor it in order to identify its roots. In this case, the polynomial factors as
[tex]$$
f(x) = (x-2)(x+3)(x+8).
$$[/tex]
This factorization immediately tells us that the function has three real roots (or zeros) given by:
[tex]$$
x=2,\quad x=-3,\quad \text{and} \quad x=-8.
$$[/tex]
These are the points where the graph crosses the [tex]$x$[/tex]‑axis.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 2. Analyzing the End Behavior
Since [tex]$f(x)$[/tex] is a cubic polynomial with a positive leading coefficient, its end behavior is as follows:
- As [tex]$x\to \infty$[/tex], [tex]$f(x)\to \infty$[/tex].
- As [tex]$x\to -\infty$[/tex], [tex]$f(x)\to -\infty$[/tex].
This means the graph falls to the left and rises to the right.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 3. Finding the Critical Points
To determine the locations of local maximum and minimum values (i.e. the turning points of the graph), we compute the derivative of [tex]$f(x)$[/tex]:
[tex]$$
f'(x)=\frac{d}{dx}\left(x^3+9x^2+2x-48\right)=3x^2+18x+2.
$$[/tex]
We then solve the equation
[tex]$$
3x^2+18x+2=0
$$[/tex]
for [tex]$x$[/tex]. The exact solutions are:
[tex]$$
x=-3-\frac{5\sqrt{3}}{3}\quad \text{and}\quad x=-3+\frac{5\sqrt{3}}{3}.
$$[/tex]
For clarity, their numerical approximations are approximately:
[tex]$$
x\approx -5.887\quad \text{and}\quad x\approx -0.113.
$$[/tex]
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 4. Classifying the Critical Points
To decide whether these critical points correspond to local maxima or minima, we examine the second derivative:
[tex]$$
f''(x)=\frac{d}{dx}(3x^2+18x+2)=6x+18.
$$[/tex]
- At [tex]$x\approx -5.887$[/tex], we have
[tex]$$
f''(-5.887) \approx 6(-5.887)+18\approx -35.322+18\approx -17.322.
$$[/tex]
Since [tex]$f''(-5.887)<0$[/tex], this critical point is a local maximum.
- At [tex]$x\approx -0.113$[/tex], we have
[tex]$$
f''(-0.113) \approx 6(-0.113)+18\approx -0.678+18\approx 17.322.
$$[/tex]
Since [tex]$f''(-0.113)>0$[/tex], this critical point is a local minimum.
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 5. Summary of the Graph’s Features
Based on the analysis, the graph of
[tex]$$
f(x)=x^3+9x^2+2x-48
$$[/tex]
has the following characteristics:
1. Zeros of the Function:
The graph crosses the [tex]$x$[/tex]-axis at
[tex]$$
x=-8,\quad x=-3,\quad \text{and} \quad x=2.
$$[/tex]
2. End Behavior:
As [tex]$x\to -\infty$[/tex], [tex]$f(x)\to -\infty$[/tex], and as [tex]$x\to \infty$[/tex], [tex]$f(x)\to \infty$[/tex].
3. Critical Points:
- A local maximum at [tex]$x\approx -5.887$[/tex], and
- A local minimum at [tex]$x\approx -0.113$[/tex].
4. Y-Intercept:
When [tex]$x=0$[/tex],
[tex]$$
f(0)=0^3+9\cdot0^2+2\cdot0-48=-48.
$$[/tex]
So the graph crosses the [tex]$y$[/tex]-axis at [tex]$(0,-48)$[/tex].
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Step 6. Describing the Graph
When you graph the function on a tool such as Desmos, you will observe:
- The curve crosses the [tex]$x$[/tex]‑axis at [tex]$x=-8$[/tex], [tex]$x=-3$[/tex], and [tex]$x=2$[/tex].
- You will notice a turning point (local maximum) near [tex]$x\approx -5.887$[/tex] and a turning point (local minimum) near [tex]$x\approx -0.113$[/tex].
- The overall shape is typical of a cubic polynomial with the left end descending to [tex]$-\infty$[/tex] and the right end rising to [tex]$\infty$[/tex].
–––––––––––––––––––––––––––––––––––––––––––––––––––––
Conclusion
The graph of
[tex]$$
f(x)=x^3+9x^2+2x-48
$$[/tex]
can be completely described by its factored form
[tex]$$
(x-2)(x+3)(x+8),
$$[/tex]
its zeros at [tex]$x=-8$[/tex], [tex]$-3$[/tex], and [tex]$2$[/tex], the end behavior as [tex]$x\to\pm\infty$[/tex], and its critical points located approximately at [tex]$x\approx -5.887$[/tex] (local maximum) and [tex]$x\approx -0.113$[/tex] (local minimum). Plotting the function on a graphing utility will confirm these features and display the characteristic cubic “S”-shape.
