What Is The \[$ X \$\]-intercept Of The Graph Of The Function \[$ F(x) = X^2 - 16x + 64 \$\]?A. \[$(0, 8)\$\] B. \[$(0, -8)\$\] C. \[$(8, 0)\$\] D. \[$(-8, 0)\$\]
In mathematics, the graph of a function is a visual representation of the relationship between the input values (x) and the output values (y). The x-intercept of a graph is the point where the graph intersects the x-axis, meaning the y-coordinate is zero. In this article, we will explore the concept of x-intercept and how to find it for a given function.
What is the x-Intercept?
The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the y-coordinate is zero, and the x-coordinate represents the value of the input that results in a zero output. In other words, the x-intercept is the solution to the equation f(x) = 0.
Finding the x-Intercept of a Quadratic Function
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (x) is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. To find the x-intercept of a quadratic function, we need to set the function equal to zero and solve for x.
The Given Function
The given function is f(x) = x^2 - 16x + 64. This is a quadratic function with a leading coefficient of 1, a linear coefficient of -16, and a constant term of 64.
Solving for the x-Intercept
To find the x-intercept of the given function, we need to set the function equal to zero and solve for x. We can do this by factoring the quadratic expression or by using the quadratic formula.
Factoring the Quadratic Expression
The quadratic expression x^2 - 16x + 64 can be factored as (x - 8)(x - 8). This means that the function can be written as f(x) = (x - 8)^2.
Setting the Function Equal to Zero
To find the x-intercept, we need to set the function equal to zero. We can do this by setting (x - 8)^2 = 0.
Solving for x
To solve for x, we can take the square root of both sides of the equation. This gives us x - 8 = 0.
Finding the x-Intercept
To find the x-intercept, we need to add 8 to both sides of the equation. This gives us x = 8.
Conclusion
In conclusion, the x-intercept of the graph of the function f(x) = x^2 - 16x + 64 is (8, 0). This means that the graph crosses the x-axis at the point (8, 0).
Answer
The correct answer is C. (8, 0).
Additional Information
The x-intercept of a graph is an important concept in mathematics, particularly in algebra and calculus. It is used to analyze the behavior of functions and to solve equations. In this article, we have explored the concept of x-intercept and how to find it for a given function. We have also seen how to use factoring and the quadratic formula to solve for the x-intercept.
Real-World Applications
The concept of x-intercept has many real-world applications. For example, in physics, the x-intercept of a graph can represent the position of an object at a given time. In economics, the x-intercept of a graph can represent the equilibrium price of a good. In engineering, the x-intercept of a graph can represent the stability of a system.
Common Mistakes
When finding the x-intercept of a graph, there are several common mistakes to avoid. One mistake is to forget to set the function equal to zero. Another mistake is to forget to solve for x. A third mistake is to forget to check the solutions for extraneous solutions.
Tips and Tricks
When finding the x-intercept of a graph, there are several tips and tricks to keep in mind. One tip is to use factoring to simplify the quadratic expression. Another tip is to use the quadratic formula to solve for x. A third tip is to check the solutions for extraneous solutions.
Conclusion
In our previous article, we explored the concept of x-intercept and how to find it for a given function. In this article, we will answer some frequently asked questions about x-intercept.
Q: What is the x-intercept of a graph?
A: The x-intercept of a graph is the point where the graph intersects the x-axis, meaning the y-coordinate is zero.
Q: How do I find the x-intercept of a graph?
A: To find the x-intercept of a graph, you need to set the function equal to zero and solve for x. You can use factoring or the quadratic formula to solve for x.
Q: What is the difference between the x-intercept and the y-intercept?
A: The x-intercept is the point where the graph intersects the x-axis, while the y-intercept is the point where the graph intersects the y-axis.
Q: Can the x-intercept be a complex number?
A: Yes, the x-intercept can be a complex number. In fact, complex numbers are often used to represent the x-intercept of a graph.
Q: How do I know if the x-intercept is real or complex?
A: To determine if the x-intercept is real or complex, you need to check the discriminant of the quadratic equation. If the discriminant is positive, the x-intercept is real. If the discriminant is negative, the x-intercept is complex.
Q: Can the x-intercept be a repeated root?
A: Yes, the x-intercept can be a repeated root. In fact, repeated roots are often used to represent the x-intercept of a graph.
Q: How do I find the x-intercept of a graph with a repeated root?
A: To find the x-intercept of a graph with a repeated root, you need to use the quadratic formula and set the discriminant equal to zero.
Q: Can the x-intercept be a rational root?
A: Yes, the x-intercept can be a rational root. In fact, rational roots are often used to represent the x-intercept of a graph.
Q: How do I find the x-intercept of a graph with a rational root?
A: To find the x-intercept of a graph with a rational root, you need to use the rational root theorem and set the function equal to zero.
Q: Can the x-intercept be a non-rational root?
A: Yes, the x-intercept can be a non-rational root. In fact, non-rational roots are often used to represent the x-intercept of a graph.
Q: How do I find the x-intercept of a graph with a non-rational root?
A: To find the x-intercept of a graph with a non-rational root, you need to use the quadratic formula and set the discriminant equal to zero.
Q: Can the x-intercept be a transcendental root?
A: Yes, the x-intercept can be a transcendental root. In fact, transcendental roots are often used to represent the x-intercept of a graph.
Q: How do I find the x-intercept of a graph with a transcendental root?
A: To find the x-intercept of a graph with a transcendental root, you need to use the quadratic formula and set the discriminant equal to zero.
Conclusion
In conclusion, the x-intercept of a graph is an important concept in mathematics. It is used to analyze the behavior of functions and to solve equations. In this article, we have answered some frequently asked questions about x-intercept and provided tips and tricks for finding the x-intercept of a graph.
Additional Resources
For more information on x-intercept, please refer to the following resources:
Common Mistakes
When finding the x-intercept of a graph, there are several common mistakes to avoid. One mistake is to forget to set the function equal to zero. Another mistake is to forget to solve for x. A third mistake is to forget to check the solutions for extraneous solutions.
Tips and Tricks
When finding the x-intercept of a graph, there are several tips and tricks to keep in mind. One tip is to use factoring to simplify the quadratic expression. Another tip is to use the quadratic formula to solve for x. A third tip is to check the solutions for extraneous solutions.
Conclusion
In conclusion, the x-intercept of a graph is an important concept in mathematics. It is used to analyze the behavior of functions and to solve equations. In this article, we have answered some frequently asked questions about x-intercept and provided tips and tricks for finding the x-intercept of a graph.