Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which graph corresponds to the function \( f(x) = 0.03x^2(x^2 - 25) \), let's break down and analyze the function step by step.
### Step 1: Simplify the Function
Rewrite the function to make it easier to analyze:
[tex]\[ f(x) = 0.03x^2(x^2 - 25) \][/tex]
[tex]\[ f(x) = 0.03x^2(x^2 - 5^2) \][/tex]
Notice that \( x^2 - 25 \) can be factored as a difference of squares:
[tex]\[ f(x) = 0.03x^2(x + 5)(x - 5) \][/tex]
### Step 2: Identify the Roots of the Function
Set the function equal to zero to find the roots:
[tex]\[ 0 = 0.03x^2(x + 5)(x - 5) \][/tex]
This equation is equal to zero if any of the factors is equal to zero:
[tex]\[ x^2 = 0 \implies x = 0 \][/tex]
[tex]\[ x + 5 = 0 \implies x = -5 \][/tex]
[tex]\[ x - 5 = 0 \implies x = 5 \][/tex]
So, the roots of the function are \( x = 0 \), \( x = -5 \), and \( x = 5 \).
### Step 3: Determine the End Behavior of the Function
The function \( f(x) \) can be expressed in terms of leading terms to determine its end behavior:
[tex]\[ f(x) = 0.03x^4 - 0.03(25)x^2 \][/tex]
[tex]\[ f(x) = 0.03x^4 - 0.75x^2 \][/tex]
The highest degree term \( 0.03x^4 \) dominates as \( x \to \pm\infty \). Because the coefficient \( 0.03 \) is positive, the function will tend to \( +\infty \) as \( x \) tends to \( \pm\infty \).
### Step 4: Analyze the Function Behavior at the Roots
Since the function is a product of polynomials:
[tex]\[ f(x) = 0.03x^2(x + 5)(x - 5) \][/tex]
At \( x = 0 \), \( x = -5 \), and \( x = 5 \), the function crosses the x-axis.
- Near \( x = 0 \), the term \( x^2 \) suggests the function behavior is quadratic around zero. That means the graph will touch the x-axis at \( x = 0 \) and be shaped like a parabola opening upwards or downwards, but in this case, opening upwards due to the overall positive coefficient when \( x \to 0. \)
- At \( x = -5 \) and \( x = 5 \), the function will cross the x-axis since it's changing signs around the roots.
### Step 5: Determine the Behavior around the Intervals
- For \( x \in (-\infty, -5) \), \( f(x) \) is positive.
- For \( x \in (-5, 0) \), \( f(x) \) is negative.
- For \( x \in (0, 5) \), \( f(x) \) is negative.
- For \( x \in (5, \infty) \), \( f(x) \) is positive.
### Summary
Using the roots and the end behavior:
- The function touches the x-axis at \( x = 0 \) but does not cross it.
- The function crosses the x-axis at \( x = -5 \) and \( x = 5 \).
- As \( x \to \pm\infty \), \( f(x) \to +\infty \).
Thus, the graph should show these behaviors: crossing the x-axis at -5 and 5, touching the x-axis at 0, and having arms heading upwards as [tex]\( x \to \pm\infty \)[/tex]. The graph with these characteristics should be the one that fits [tex]\( f(x) = 0.03x^2(x^2 - 25) \)[/tex].
### Step 1: Simplify the Function
Rewrite the function to make it easier to analyze:
[tex]\[ f(x) = 0.03x^2(x^2 - 25) \][/tex]
[tex]\[ f(x) = 0.03x^2(x^2 - 5^2) \][/tex]
Notice that \( x^2 - 25 \) can be factored as a difference of squares:
[tex]\[ f(x) = 0.03x^2(x + 5)(x - 5) \][/tex]
### Step 2: Identify the Roots of the Function
Set the function equal to zero to find the roots:
[tex]\[ 0 = 0.03x^2(x + 5)(x - 5) \][/tex]
This equation is equal to zero if any of the factors is equal to zero:
[tex]\[ x^2 = 0 \implies x = 0 \][/tex]
[tex]\[ x + 5 = 0 \implies x = -5 \][/tex]
[tex]\[ x - 5 = 0 \implies x = 5 \][/tex]
So, the roots of the function are \( x = 0 \), \( x = -5 \), and \( x = 5 \).
### Step 3: Determine the End Behavior of the Function
The function \( f(x) \) can be expressed in terms of leading terms to determine its end behavior:
[tex]\[ f(x) = 0.03x^4 - 0.03(25)x^2 \][/tex]
[tex]\[ f(x) = 0.03x^4 - 0.75x^2 \][/tex]
The highest degree term \( 0.03x^4 \) dominates as \( x \to \pm\infty \). Because the coefficient \( 0.03 \) is positive, the function will tend to \( +\infty \) as \( x \) tends to \( \pm\infty \).
### Step 4: Analyze the Function Behavior at the Roots
Since the function is a product of polynomials:
[tex]\[ f(x) = 0.03x^2(x + 5)(x - 5) \][/tex]
At \( x = 0 \), \( x = -5 \), and \( x = 5 \), the function crosses the x-axis.
- Near \( x = 0 \), the term \( x^2 \) suggests the function behavior is quadratic around zero. That means the graph will touch the x-axis at \( x = 0 \) and be shaped like a parabola opening upwards or downwards, but in this case, opening upwards due to the overall positive coefficient when \( x \to 0. \)
- At \( x = -5 \) and \( x = 5 \), the function will cross the x-axis since it's changing signs around the roots.
### Step 5: Determine the Behavior around the Intervals
- For \( x \in (-\infty, -5) \), \( f(x) \) is positive.
- For \( x \in (-5, 0) \), \( f(x) \) is negative.
- For \( x \in (0, 5) \), \( f(x) \) is negative.
- For \( x \in (5, \infty) \), \( f(x) \) is positive.
### Summary
Using the roots and the end behavior:
- The function touches the x-axis at \( x = 0 \) but does not cross it.
- The function crosses the x-axis at \( x = -5 \) and \( x = 5 \).
- As \( x \to \pm\infty \), \( f(x) \to +\infty \).
Thus, the graph should show these behaviors: crossing the x-axis at -5 and 5, touching the x-axis at 0, and having arms heading upwards as [tex]\( x \to \pm\infty \)[/tex]. The graph with these characteristics should be the one that fits [tex]\( f(x) = 0.03x^2(x^2 - 25) \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
