f(x) = -4(x+2)(x - 5)^3
- 2, multiplicity 1, touches the x-axis and turns around; 5, multiplicity 3, touches x-axis and turns around
I left (x+2) out of the equation. Can you explain to me these type of problems, I only understand a little bit of it.
You have it right, if another x+2 is in the equation twice. Your understanding of odd and even funcions, and roots, is ok.
Certainly! Let's break down the given equation and understand the concepts involved.
First, let's analyze the equation: f(x) = -4(x+2)(x - 5)^3.
The equation represents a polynomial function, where f(x) is the output (or y-value) corresponding to the input (or x-value) of x.
In this case, it appears to be a cubic function with two factors: (x+2) and (x - 5)^3. The factor (x - 5)^3 indicates that the value 5 has a multiplicity of 3.
Now, let's understand what the terms "multiplicity" and "touches x-axis and turns around" mean in the context of this equation.
1. Multiplicity: In the equation, when a factor or root of the function appears multiple times, it is called multiplicity. It gives us information about how the graph of the function behaves near that particular root.
2. Touches x-axis and turns around: When a root, or x-intercept, of the function touches the x-axis and changes direction, it indicates a turning point in the graph of the function.
In the given equation, we have two roots: -2 and 5.
For -2:
- Multiplicity of 1 means that -2 is a simple root and appears only once.
- It touches the x-axis and turns around, which means the graph will cross the x-axis at -2 and change direction.
For 5:
- Multiplicity of 3 means that 5 is a triple root and appears three times.
- It also touches the x-axis and turns around, indicating that the graph will cross the x-axis at 5 and change direction three times.
To understand the behavior of the graph of this equation more accurately, you can graph it using a graphing calculator or software. This will give you a visual representation of how the function behaves around the roots and turning points.
I hope this explanation helps! Let me know if you have any further questions.