To determine the graph that matches the function [tex]\( f(x) = (x+3)x \)[/tex], we need to analyze the zeros of the function and the end behavior of its graph. Let's go through this in detail:
### 1. Finding the Zeros of the Function:
The zeros of a function are the values of [tex]\( x \)[/tex] at which [tex]\( f(x) = 0 \)[/tex].
For the function [tex]\( f(x) = (x+3)x \)[/tex]:
1. Expand the function:
[tex]\[
f(x) = (x + 3)x = x^2 + 3x
\][/tex]
2. Set the function equal to zero to find the zeros:
[tex]\[
x^2 + 3x = 0
\][/tex]
3. Factor out [tex]\( x \)[/tex]:
[tex]\[
x(x + 3) = 0
\][/tex]
4. Solve for the values of [tex]\( x \)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad x + 3 = 0
\][/tex]
[tex]\[
x = 0 \quad \text{or} \quad x = -3
\][/tex]
Therefore, the function has two zeros: [tex]\( x = 0 \)[/tex] and [tex]\( x = -3 \)[/tex].
### 2. Determining the End Behavior of the Graph:
To understand the end behavior, observe what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity.
For the function [tex]\( f(x) = x^2 + 3x \)[/tex]:
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]):
[tex]\[
f(x) = x^2 + 3x \to \infty
\][/tex]
Since the [tex]\( x^2 \)[/tex] term dominates, [tex]\( f(x) \)[/tex] grows without bound as [tex]\( x \)[/tex] becomes very large in the positive direction.
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
[tex]\[
f(x) = x^2 + 3x \to \infty
\][/tex]
Although [tex]\( 3x \)[/tex] is negative for large negative [tex]\( x \)[/tex], the [tex]\( x^2 \)[/tex] term again dominates, and since it is always positive, [tex]\( f(x) \)[/tex] also grows without bound as [tex]\( x \)[/tex] becomes very large in the negative direction.
### Conclusion:
- The function [tex]\( f(x) \)[/tex] has zeros at [tex]\( x = 0 \)[/tex] and [tex]\( x = -3 \)[/tex].
- The end behavior of the function indicates that 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].
With these observations, the graph of the function [tex]\( f(x) = (x+3)x \)[/tex] should have the following characteristics:
- It intersects the x-axis at [tex]\( x = 0 \)[/tex] and [tex]\( x = -3 \)[/tex].
- The graph opens upwards on both ends, indicating that it rises to infinity as [tex]\( x \)[/tex] moves towards both positive and negative infinity.
By ensuring these characteristics match, you can confidently select the corresponding graph.
