Get the best answers to your questions with the help of IDNLearn.com's experts. Get thorough and trustworthy answers to your queries from our extensive network of knowledgeable professionals.
Sagot :
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.
### 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.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.
