Find The $x$-intercepts Of The Polynomial Function. State Whether The Graph Crosses The $x$-axis, Or Touches The $x$-axis And Turns Around, At Each Intercept.Given Function: $f(x) = 4x^2 - X^3$A. 0, Crosses

Introduction

In mathematics, the $x$-intercepts of a polynomial function are the points where the graph of the function crosses or touches the $x$-axis. These points are also known as the roots or zeros of the function. In this article, we will find the $x$-intercepts of the given polynomial function $f(x) = 4x^2 - x^3$ and determine whether the graph crosses the $x$-axis or touches the $x$-axis and turns around at each intercept.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$, where $a = -1$, $b = 0$, and $c = 4$. However, we can rewrite the function as $f(x) = -x^3 + 4x^2$ by factoring out a negative sign from the first term.

Finding the $x$-Intercepts

To find the $x$-intercepts of the function, we need to set the function equal to zero and solve for $x$. This is because the $x$-intercepts are the points where the graph of the function crosses the $x$-axis, and at these points, the value of the function is zero.

$$f(x) = 0$$

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

Factoring the Equation

We can factor out an $x^2$ term from the equation:

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

Solving for $x$

To find the values of $x$ that satisfy the equation, we need to set each factor equal to zero and solve for $x$.

$$x^2 = 0$$

$$-x + 4 = 0$$

Solving the First Equation

The first equation is a quadratic equation in the form of $x^2 = 0$. This equation has a repeated root at $x = 0$.

Solving the Second Equation

The second equation is a linear equation in the form of $-x + 4 = 0$. We can solve for $x$ by adding $x$ to both sides of the equation and then dividing both sides by $-1$.

$$-x + 4 = 0$$

$$-x = -4$$

$$x = 4$$

Determining the Behavior of the Graph

Now that we have found the $x$-intercepts of the function, we need to determine whether the graph crosses the $x$-axis or touches the $x$-axis and turns around at each intercept.

The graph of the function $f(x) = -x^3 + 4x^2$ is a cubic function, which means that it can have at most two turning points. The graph of the function is a downward-facing parabola, which means that it opens downward.

Conclusion

In conclusion, the $x$-intercepts of the polynomial function $f(x) = 4x^2 - x^3$ are $x = 0$ and $x = 4$. The graph of the function crosses the $x$-axis at $x = 0$ and touches the $x$-axis and turns around at $x = 4$.

Final Answer

The final answer is:

• x = 0$, crosses

• x = 4$, touches and turns around

Discussion

The discussion category for this article is mathematics. The article provides a step-by-step solution to finding the $x$-intercepts of a polynomial function and determining the behavior of the graph at each intercept. The article is written in a clear and concise manner, making it easy for readers to understand the concepts and follow the solution.

Related Articles

• Finding the $x$-Intercepts of a Quadratic Function
• Determining the Behavior of a Graph at an $x$-Intercept
• Solving Polynomial Equations

Keywords

• x$-intercepts

• polynomial function
• cubic function
• quadratic function
• roots
• zeros
• turning points
• graphing
• mathematics
  

Introduction

In our previous article, we discussed how to find the $x$-intercepts of a polynomial function and determine the behavior of the graph at each intercept. In this article, we will answer some frequently asked questions related to finding the $x$-intercepts of a polynomial function.

Q1: What is the difference between a root and a zero of a polynomial function?

A1: A root and a zero of a polynomial function are the same thing. They refer to the values of $x$ that make the function equal to zero. The terms "root" and "zero" are often used interchangeably in mathematics.

Q2: How do I find the $x$-intercepts of a polynomial function?

A2: To find the $x$-intercepts of a polynomial function, you need to set the function equal to zero and solve for $x$. This will give you the values of $x$ where the graph of the function crosses the $x$-axis.

Q3: What is the relationship between the $x$-intercepts of a polynomial function and its graph?

A3: The $x$-intercepts of a polynomial function are the points where the graph of the function crosses the $x$-axis. The graph of the function may cross the $x$-axis at one or more points, or it may touch the $x$-axis and turn around at one or more points.

Q4: How do I determine whether the graph of a polynomial function crosses the $x$-axis or touches the $x$-axis and turns around at an $x$-intercept?

A4: To determine whether the graph of a polynomial function crosses the $x$-axis or touches the $x$-axis and turns around at an $x$-intercept, you need to examine the behavior of the function at that point. If the function changes from increasing to decreasing or from decreasing to increasing at the $x$-intercept, then the graph touches the $x$-axis and turns around at that point. If the function does not change from increasing to decreasing or from decreasing to increasing at the $x$-intercept, then the graph crosses the $x$-axis at that point.

Q5: Can a polynomial function have more than two $x$-intercepts?

A5: No, a polynomial function can have at most two $x$-intercepts. This is because a polynomial function of degree $n$ can have at most $n$ turning points, and the $x$-intercepts are the points where the graph of the function crosses or touches the $x$-axis.

Q6: How do I find the $x$-intercepts of a polynomial function with complex coefficients?

A6: To find the $x$-intercepts of a polynomial function with complex coefficients, you need to use the quadratic formula or other methods to solve the equation. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the polynomial function.

Q7: Can a polynomial function have an $x$-intercept at $x = 0$?

A7: Yes, a polynomial function can have an $x$-intercept at $x = 0$. This is because the value of the function at $x = 0$ is zero, which means that the graph of the function crosses the $x$-axis at $x = 0$.

Q8: How do I determine whether the graph of a polynomial function is a parabola or a cubic curve?

A8: To determine whether the graph of a polynomial function is a parabola or a cubic curve, you need to examine the degree of the polynomial function. If the degree of the polynomial function is 2, then the graph is a parabola. If the degree of the polynomial function is 3, then the graph is a cubic curve.

Conclusion

In conclusion, finding the $x$-intercepts of a polynomial function is an important concept in mathematics. By understanding how to find the $x$-intercepts of a polynomial function and determine the behavior of the graph at each intercept, you can gain a deeper understanding of the properties of polynomial functions and their graphs.

Final Answer

The final answer is:

• The $x$-intercepts of a polynomial function are the points where the graph of the function crosses or touches the $x$-axis.
• The graph of a polynomial function may cross the $x$-axis at one or more points, or it may touch the $x$-axis and turn around at one or more points.
• The degree of a polynomial function determines the shape of its graph.

Discussion

The discussion category for this article is mathematics. The article provides a step-by-step solution to finding the $x$-intercepts of a polynomial function and determining the behavior of the graph at each intercept. The article is written in a clear and concise manner, making it easy for readers to understand the concepts and follow the solution.

Related Articles

• Finding the $x$-Intercepts of a Quadratic Function
• Determining the Behavior of a Graph at an $x$-Intercept
• Solving Polynomial Equations

Keywords

• x$-intercepts

• polynomial function
• cubic function
• quadratic function
• roots
• zeros
• turning points
• graphing
• mathematics