Question

2 use a graphing calculator to graph the following function. $f(x) = x^3 - 12x + 16$. then, identify the characteristics of the functi the domain is the range is the x-intercept is the y-intercept is interval increasing interval decreasing positive interval negative interval

Explanation:

Step1: Find domain

Polynomial functions have domain \(\mathbb{R}\).

Step2: Find range

Cubic polynomial (odd degree) has range \(\mathbb{R}\).

Step3: Find x-intercepts

Set \(f(x)=0\): \(x^3 -12x +16=(x-2)^2(x+4)=0\) → \(x=2\), \(x=-4\).

Step4: Find y-intercept

Set \(x=0\): \(f(0)=0 -0 +16=16\).

Step5: Find critical points

Derivative \(f'(x)=3x^2 -12\). Set \(f'(x)=0\): \(3(x^2 -4)=0\) → \(x=\pm2\).

Step6: Determine increasing/decreasing

• \(x<-2\) or \(x>2\): \(f'(x)>0\) (increasing)
• \(-2

Step7: Determine positive/negative

• \(x<-4\): \(f(x)<0\)
• \(-4
• \(x>2\): \(f(x)>0\) (since \((x-2)^2\geq0\) and \(x+4>0\))

Step8: Confirm with graph

Graphing calculator verifies the above results.

Answer:

Domain: All real numbers
Range: All real numbers
x-intercepts: \(x = 2\), \(x = -4\)
y-intercept: \(16\)
Increasing intervals: \((-\infty, -2)\) and \((2, \infty)\)
Decreasing interval: \((-2, 2)\)
Positive intervals: \((-4, 2)\) and \((2, \infty)\)
Negative interval: \((-\infty, -4)\)