To identify which graph corresponds to the function [tex]\( f(x) = -0.08 x (x^2 - 11x + 18) \)[/tex], we can analyze the function step-by-step:
### Step 1: Expand the Function
First, let's expand the function:
[tex]\[ f(x) = -0.08 x (x^2 - 11x + 18) \][/tex]
Distribute [tex]\( -0.08x \)[/tex] across the terms inside the parentheses:
[tex]\[ f(x) = -0.08 x^3 + 0.88 x^2 - 1.44 x \][/tex]
### Step 2: Identify the Key Characteristics
Next, we analyze the critical points and behavior of the function.
1. Roots or Zeros:
We need to find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -0.08 x (x^2 - 11x + 18) = 0 \][/tex]
This product is zero if either factor is zero:
[tex]\[ -0.08x = 0 \quad \text{or} \quad x^2 - 11x + 18 = 0 \][/tex]
Solving for the linear factor:
[tex]\[ x = 0 \][/tex]
Solving the quadratic equation:
[tex]\[ x^2 - 11x + 18 = 0 \][/tex]
[tex]\[ (x - 2)(x - 9) = 0 \][/tex]
[tex]\[ x = 2 \quad \text{or} \quad x = 9 \][/tex]
Therefore, the roots of the function are [tex]\( x = 0 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 9 \)[/tex].
2. Behavior of the Function:
Since the leading term is negative ([tex]\( -0.08x^3 \)[/tex]), the function [tex]\( f(x) \)[/tex] is a cubic polynomial that points downward as [tex]\( x \)[/tex] moves towards positive or negative infinity.
3. Additional Key Points:
To understand more, we can check the function’s value at several other points to gauge its shape. However, we already have crucial information based on the zeros and the leading coefficient.
### Step 3: Putting it All Together
Given the roots and behavior of the function, we can sketch the graph:
- The graph has x-intercepts at [tex]\( x = 0 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 9 \)[/tex].
- Since it is a cubic polynomial with a negative leading term, the graph starts from positive infinity, crosses the x-axis at [tex]\( x = 9 \)[/tex], goes down to a local minimum or maximum, crosses the x-axis again at [tex]\( x = 2 \)[/tex], goes up to another local extremum, crosses the x-axis at [tex]\( x = 0 \)[/tex], and then falls to negative infinity.
### Conclusion
The correct graph for the function [tex]\( f(x) = -0.08 x (x^2 - 11x + 18) \)[/tex] should reflect these key points:
- Three roots at [tex]\( x = 0 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 9 \)[/tex].
- Downward behavior as [tex]\( x \)[/tex] approaches infinity in both directions.
- The general cubic function shape, with turning points reflecting the polynomial's characteristics.
By matching the described characteristics and behavior with the provided graphs, you can identify the correct graph representing [tex]\( f(x) = -0.08 x (x^2 - 11x + 18) \)[/tex].
