Answer :
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].
