Pregunta
For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through \( e \). A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are 0,2 . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and turn around? Select the correct hoice below and, if necessary, fill in the answer box to complete your choice A. The \( x \)-intercept(s) at which the graph touches the \( x \)-axis and turns around is/are \( \square \) I. (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c. Find the \( y \)-intercept. The \( y \)-intercept is 0. (Simplify your answer. Type an integer or a decimal.) d. Determine whether the graph has \( y \)-axis symmetry, origin symmetry, or neither. Choose the correct answer below. A. The graph of f is symmetric about the origin. B. The graph of \( f \) is symmetric about the \( y \)-axis. C. The graph of \( f \) is neither symmetric about the \( y \)-axis nor symmetric about the origin.
Ask by Flynn Wheeler. in the United States
Mar 19,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
**a. \( x \)-intercepts where the graph crosses the \( x \)-axis:**
0, 2
**b. \( x \)-intercepts where the graph touches and turns around:**
None
**c. \( y \)-intercept:**
0
**d. Symmetry of the Graph:**
Neither symmetric about the \( y \)-axis nor about the origin.
Solución
**a. \( x \)-intercepts where the graph crosses the \( x \)-axis**
Factor the function:
\[
f(x) = -4x^4 + 8x^3 = -4x^3(x-2)
\]
Set \( f(x) = 0 \):
\[
-4x^3(x-2)=0
\]
The solutions are:
\[
x^3 = 0 \quad \Rightarrow \quad x=0
\]
\[
x-2=0 \quad \Rightarrow \quad x=2
\]
Thus, the \( x \)-intercepts at which the graph crosses the \( x \)-axis are:
\[
0,2
\]
---
**b. \( x \)-intercepts where the graph touches and turns around**
The behavior at an \( x \)-intercept is determined by the multiplicity of the factor.
- The factor \( x^3 \) for \( x=0 \) has multiplicity 3 (an odd multiplicity), so the graph crosses the \( x \)-axis at \( x=0 \).
- The factor \( (x-2) \) has multiplicity 1 (also odd), so the graph crosses the \( x \)-axis at \( x=2 \).
Thus, there are no \( x \)-intercepts at which the graph touches and turns around.
---
**c. \( y \)-intercept**
To find the \( y \)-intercept, evaluate \( f(0) \):
\[
f(0)=-4(0)^4+8(0)^3=0
\]
The \( y \)-intercept is:
\[
0
\]
---
**d. Symmetry of the Graph**
Check \( f(-x) \):
\[
f(-x)=-4(-x)^4+8(-x)^3=-4x^4-8x^3
\]
This is neither equal to \( f(x)=-4x^4+8x^3 \) nor to \( -f(x)=4x^4-8x^3 \).
Thus, the graph is neither symmetric about the \( y \)-axis nor about the origin.
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Simplifique esta solución Beyond the Answer
The graph of the polynomial function \( f(x) = -4x^4 + 8x^3 \) has some interesting features! The polynomial has a degree of 4, which means it can have up to 4 real roots. Upon factoring, we find that the \( x \)-intercepts where the graph crosses the \( x \)-axis indeed occur at \( x = 0 \) and \( x = 2 \). These represent the points where the graph intersects the axis, providing a nice visual cue of the function's behavior! However, there's also the \( x \)-intercept where the graph touches the \( x \)-axis. This happens at \( x = 0 \), indicating that it's a double root (as seen in the factored form). Here, the graph bounces off the axis, showcasing an essential characteristic of polynomial functions.
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