Which Statement Describes The Graph Of F ( X ) = − X 4 + 3 X 3 + 10 X 2 F(x) = -x^4 + 3x^3 + 10x^2 F ( X ) = − X 4 + 3 X 3 + 10 X 2 ?A. The Graph Crosses The X X X -axis At X = 0 X = 0 X = 0 And Touches The X X X -axis At X = 5 X = 5 X = 5 And X = − 2 X = -2 X = − 2 .B. The Graph Touches The

Mar 8, 2025 by ADMIN 293 views

**Analyzing the Graph of a Polynomial Function**

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Understanding the Problem

The given problem involves analyzing the graph of a polynomial function, specifically $f(x) = -x^4 + 3x^3 + 10x^2$ . We are asked to determine which statement accurately describes the graph of this function.

Key Concepts

Before diving into the analysis, it's essential to understand the key concepts involved:

Graph of a polynomial function: The graph of a polynomial function is a visual representation of the function's behavior, including its shape, intercepts, and turning points.
x-intercepts: The x-intercepts of a graph are the points where the graph crosses the x-axis, i.e., where the function value is zero.
Turning points: The turning points of a graph are the points where the function changes from increasing to decreasing or vice versa.

Analyzing the Function

Let's start by analyzing the given function:

$f(x) = -x^4 + 3x^3 + 10x^2$

We can see that the function is a quartic polynomial, meaning it has a degree of 4. This implies that the graph of the function will have at most 3 turning points.

Finding the x-Intercepts

To find the x-intercepts of the graph, we need to set the function value equal to zero and solve for x:

$-x^4 + 3x^3 + 10x^2 = 0$

We can factor out an $x^2$ term:

$x^2(-x^2 + 3x + 10) = 0$

This gives us two possible solutions:

$x^2 = 0$ or $-x^2 + 3x + 10 = 0$

Solving the first equation, we get:

$x^2 = 0 \Rightarrow x = 0$

This is the only x-intercept of the graph.

Finding the Turning Points

To find the turning points of the graph, we need to find the critical points of the function. Critical points occur when the derivative of the function is equal to zero or undefined.

Let's find the derivative of the function:

$f'(x) = -4x^3 + 9x^2 + 20x$

Now, we need to set the derivative equal to zero and solve for x:

$-4x^3 + 9x^2 + 20x = 0$

We can factor out an $x$ term:

$x(-4x^2 + 9x + 20) = 0$

This gives us two possible solutions:

$x = 0$ or $-4x^2 + 9x + 20 = 0$

Solving the first equation, we get:

$x = 0$

This is the only critical point of the graph.

Analyzing the Critical Points

Now that we have found the critical points, let's analyze them:

x = 0: This is the only critical point of the graph. Since it's also an x-intercept, we can conclude that the graph touches the x-axis at x = 0.

Conclusion

Based on our analysis, we can conclude that the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ touches the x-axis at x = 0.

Answer

The correct answer is:

A. The graph crosses the x-axis at x = 0 and touches the x-axis at x = 5 and x = -2.

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Q: What is the degree of the polynomial function $f(x) = -x^4 + 3x^3 + 10x^2$ ?

A: The degree of the polynomial function $f(x) = -x^4 + 3x^3 + 10x^2$ is 4, since the highest power of x is 4.

Q: How many turning points can the graph of a polynomial function have?

A: The graph of a polynomial function can have at most n-1 turning points, where n is the degree of the polynomial. In this case, the graph can have at most 3 turning points.

Q: What is the x-intercept of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ ?

A: The x-intercept of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ is x = 0.

Q: What is the critical point of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ ?

A: The critical point of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ is x = 0.

Q: Does the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ touch the x-axis at x = 5 and x = -2?

A: No, the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ does not touch the x-axis at x = 5 and x = -2. It only touches the x-axis at x = 0.

Q: What is the shape of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ ?

A: The shape of the graph of the function $f(x) = -x^4 + 3x^3 + 10x^2$ is a quartic curve that opens downward.

Q: Can the graph of a polynomial function have more than one x-intercept?

A: Yes, the graph of a polynomial function can have more than one x-intercept.

Q: Can the graph of a polynomial function have more than one turning point?

A: Yes, the graph of a polynomial function can have more than one turning point.

Q: How can I determine the number of turning points of a polynomial function?

A: You can determine the number of turning points of a polynomial function by finding the degree of the polynomial and subtracting 1.

Q: How can I find the x-intercepts of a polynomial function?

A: You can find the x-intercepts of a polynomial function by setting the function value equal to zero and solving for x.

Q: How can I find the critical points of a polynomial function?

A: You can find the critical points of a polynomial function by finding the derivative of the function and setting it equal to zero.

Q: What is the significance of the critical points of a polynomial function?

A: The critical points of a polynomial function are the points where the function changes from increasing to decreasing or vice versa.

Q: Can the graph of a polynomial function have a horizontal asymptote?

A: Yes, the graph of a polynomial function can have a horizontal asymptote.

Q: Can the graph of a polynomial function have a vertical asymptote?

A: No, the graph of a polynomial function cannot have a vertical asymptote.

Q: How can I determine the horizontal asymptote of a polynomial function?

A: You can determine the horizontal asymptote of a polynomial function by looking at the degree of the polynomial and the leading coefficient.

Q: How can I determine the vertical asymptote of a polynomial function?

A: You cannot determine the vertical asymptote of a polynomial function, as it does not exist.

Q: What is the significance of the horizontal asymptote of a polynomial function?

A: The horizontal asymptote of a polynomial function represents the behavior of the function as x approaches infinity.

Q: What is the significance of the vertical asymptote of a polynomial function?

A: There is no significance of the vertical asymptote of a polynomial function, as it does not exist.

Q: Can the graph of a polynomial function have a slant asymptote?

A: Yes, the graph of a polynomial function can have a slant asymptote.

Q: How can I determine the slant asymptote of a polynomial function?

A: You can determine the slant asymptote of a polynomial function by dividing the polynomial by the leading term and looking at the quotient.

Q: What is the significance of the slant asymptote of a polynomial function?

A: The slant asymptote of a polynomial function represents the behavior of the function as x approaches infinity.

Q: Can the graph of a polynomial function have a hole?

A: Yes, the graph of a polynomial function can have a hole.

Q: How can I determine the hole of a polynomial function?

A: You can determine the hole of a polynomial function by finding the factors of the polynomial and looking for any common factors.

Q: What is the significance of the hole of a polynomial function?

A: The hole of a polynomial function represents a point where the function is not defined.

Q: Can the graph of a polynomial function have a cusp?

A: Yes, the graph of a polynomial function can have a cusp.

Q: How can I determine the cusp of a polynomial function?

A: You can determine the cusp of a polynomial function by finding the second derivative of the function and looking for any points where the second derivative is zero.

Q: What is the significance of the cusp of a polynomial function?

A: The cusp of a polynomial function represents a point where the function changes from concave up to concave down or vice versa.

Q: Can the graph of a polynomial function have a point of inflection?

A: Yes, the graph of a polynomial function can have a point of inflection.

Q: How can I determine the point of inflection of a polynomial function?

A: You can determine the point of inflection of a polynomial function by finding the second derivative of the function and looking for any points where the second derivative is zero.

Q: What is the significance of the point of inflection of a polynomial function?

A: The point of inflection of a polynomial function represents a point where the function changes from concave up to concave down or vice versa.

Q: Can the graph of a polynomial function have a local maximum or minimum?

A: Yes, the graph of a polynomial function can have a local maximum or minimum.

Q: How can I determine the local maximum or minimum of a polynomial function?

A: You can determine the local maximum or minimum of a polynomial function by finding the critical points of the function and looking for any points where the function changes from increasing to decreasing or vice versa.

Q: What is the significance of the local maximum or minimum of a polynomial function?

A: The local maximum or minimum of a polynomial function represents a point where the function has a maximum or minimum value.

Q: Can the graph of a polynomial function have a global maximum or minimum?

A: Yes, the graph of a polynomial function can have a global maximum or minimum.

Q: How can I determine the global maximum or minimum of a polynomial function?

A: You can determine the global maximum or minimum of a polynomial function by looking at the behavior of the function as x approaches infinity.

Q: What is the significance of the global maximum or minimum of a polynomial function?

A: The global maximum or minimum of a polynomial function represents the maximum or minimum value of the function over its entire domain.

Q: Can the graph of a polynomial function have a saddle point?

A: Yes, the graph of a polynomial function can have a saddle point.

Q: How can I determine the saddle point of a polynomial function?

A: You can determine the saddle point of a polynomial function by finding the critical points of the function and looking for any points where the function changes from concave up to concave down or vice versa.

Q: What is the significance of the saddle point of a polynomial function?

A: The saddle point of a polynomial function represents a point where the function has a maximum value in one direction and a minimum value in another direction.

Q: Can the graph of a polynomial function have a col?

A: Yes, the graph of a polynomial function can have a col.

Q: How can I determine the col of a polynomial function?

A: You can determine the col of a polynomial function by finding the critical points of the function