Given The Function F ( X ) = ( X − 5 ) ( X + 6 ) ( X − 3 F(x)=(x-5)(x+6)(x-3 F ( X ) = ( X − 5 ) ( X + 6 ) ( X − 3 ]:- The F F F -intercept Is ( 0 , 90 (0, 90 ( 0 , 90 ].- The X X X -intercepts Are ( 5 , 0 (5, 0 ( 5 , 0 ], ( − 6 , 0 (-6, 0 ( − 6 , 0 ], And ( 3 , 0 (3, 0 ( 3 , 0 ].

Introduction

In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The graph of a polynomial function can be used to visualize the behavior of the function and to identify its key features, such as its x-intercepts and y-intercept. In this article, we will discuss the graph of a polynomial function and how to identify its key features.

The Function

The function we will be discussing is given by:

$$f(x) = (x-5)(x+6)(x-3)$$

This function is a cubic polynomial, meaning that it has a degree of 3. The graph of this function will be a cubic curve that passes through the points (5, 0), (-6, 0), and (3, 0).

The $f$-intercept

The $f$-intercept is the point on the graph where the function intersects the y-axis. In other words, it is the point where x = 0. To find the $f$-intercept, we can plug x = 0 into the function:

$$f(0) = (0-5)(0+6)(0-3)$$

$$f(0) = (-5)(6)(-3)$$

$$f(0) = 90$$

Therefore, the $f$-intercept is (0, 90).

The $x$-intercepts

The $x$-intercepts are the points on the graph where the function intersects the x-axis. In other words, they are the points where y = 0. To find the $x$-intercepts, we can set y = 0 and solve for x:

$$f(x) = (x-5)(x+6)(x-3) = 0$$

This equation can be factored as:

$$(x-5)(x+6)(x-3) = 0$$

This tells us that either (x-5) = 0, (x+6) = 0, or (x-3) = 0. Solving for x in each of these equations, we get:

$$x-5 = 0 \Rightarrow x = 5$$

$$x+6 = 0 \Rightarrow x = -6$$

$$x-3 = 0 \Rightarrow x = 3$$

Therefore, the $x$-intercepts are (5, 0), (-6, 0), and (3, 0).

Graphing the Function

To graph the function, we can use the information we have gathered about its key features. We know that the function has an $f$-intercept at (0, 90) and $x$-intercepts at (5, 0), (-6, 0), and (3, 0). We can use this information to sketch the graph of the function.

Conclusion

In this article, we have discussed the graph of a polynomial function and how to identify its key features, such as its $f$-intercept and $x$-intercepts. We have used the function $f(x) = (x-5)(x+6)(x-3)$ as an example and have shown how to find its $f$-intercept and $x$-intercepts. We have also sketched the graph of the function using this information.

Key Takeaways

• The graph of a polynomial function can be used to visualize the behavior of the function and to identify its key features.
• The $f$-intercept is the point on the graph where the function intersects the y-axis.
• The $x$-intercepts are the points on the graph where the function intersects the x-axis.
• To graph a polynomial function, we can use the information we have gathered about its key features.

Further Reading

For further reading on the topic of polynomial functions, we recommend the following resources:

• Wikipedia: Polynomial
• Math Is Fun: Polynomial Functions
• Khan Academy: Polynomial Functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Q&A: Understanding the Graph of a Polynomial Function =====================================================

Introduction

In our previous article, we discussed the graph of a polynomial function and how to identify its key features, such as its $f$-intercept and $x$-intercepts. In this article, we will answer some common questions about polynomial functions and their graphs.

Q: What is a polynomial function?

A: A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of the variable in the function. For example, the degree of the function $f(x) = x^3 + 2x^2 + 3x + 1$ is 3.

Q: How do I find the $f$-intercept of a polynomial function?

A: To find the $f$-intercept of a polynomial function, you can plug x = 0 into the function and solve for y.

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

A: To find the $x$-intercepts of a polynomial function, you can set y = 0 and solve for x.

Q: What is the difference between a linear function and a polynomial function?

A: A linear function is a function that can be expressed as a single term, where the variable is raised to the power of 1. A polynomial function, on the other hand, is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

Q: Can a polynomial function have more than one $f$-intercept?

A: No, a polynomial function can only have one $f$-intercept.

Q: Can a polynomial function have more than one $x$-intercept?

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

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you can use the information you have gathered about its key features, such as its $f$-intercept and $x$-intercepts.

Q: What are some common types of polynomial functions?

A: Some common types of polynomial functions include:

• Linear functions: $f(x) = ax + b$
• Quadratic functions: $f(x) = ax^2 + bx + c$
• Cubic functions: $f(x) = ax^3 + bx^2 + cx + d$
• Quartic functions: $f(x) = ax^4 + bx^3 + cx^2 + dx + e$

Q: How do I determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you can look at the highest power of the variable in the function.

Q: Can a polynomial function have a negative degree?

A: No, a polynomial function cannot have a negative degree.

Conclusion

In this article, we have answered some common questions about polynomial functions and their graphs. We hope that this information has been helpful in understanding the graph of a polynomial function.

Key Takeaways

• A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.
• The degree of a polynomial function is the highest power of the variable in the function.
• To find the $f$-intercept of a polynomial function, you can plug x = 0 into the function and solve for y.
• To find the $x$-intercepts of a polynomial function, you can set y = 0 and solve for x.

Further Reading

For further reading on the topic of polynomial functions, we recommend the following resources:

• Wikipedia: Polynomial
• Math Is Fun: Polynomial Functions
• Khan Academy: Polynomial Functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton