Question

Graph the function f(x) = x3 – 10x2 + 28x – 24 Then, write a

written description of the graph that includes all key characteristics below.
- Type of polynomial function and degree
- Domain & Range
- End behavior
- Intervals of Increasing/Decreasing
- Intervals of Positive/Negative
- Turning points (including whether it's a relative max or relative min)
- Zeros and their multiplicity (and the effect of the multiplicity on the graph)

Answer 1

- Cubic Function | 3-degree

- Domain & Range | Domain and Range both are all set of Real Numbers.

- End Behavior | When x approaches positive infinity, f(x) approaches positive infinity as well. When x approaches negative infinity, f(x) approaches negative infinity too.

- Intervals of Increasing/Decreasing | Increases at (-∞, 2) U (6,∞) and Decreases at (2,14/3) aka 2 < x < 14/3

- Intervals of Positive/Negative

Positive when x > 6 or (6,+∞)

Negative when 2 < x < 6 and x < 2 or (2,6) U (-∞,2)

- Turning Points are (2,0) and (14/3, -256/27)

- Zeros and Multiplicity | Factoring the polynomial as we get (x-2)(x-2)(x-6). Therefore, the zeros are multiplication of 2 and 6.

Multiplicity will affect the graph for x-axis. When the multiplicity kicks in, the graph will turn either decreasing on increasing on x-axis immediately.

Also I was plotting the graph so It might take too long.