What Is The $x$-intercept Of The Graph Of The Function $f(x)=x^2-16x+64$?A. $ ( − 8 , 0 ) (-8,0) ( − 8 , 0 ) [/tex] B. $(0,8)$ C. $(8,0)$ D. $ ( 0 , − 8 ) (0,-8) ( 0 , − 8 ) [/tex]
Understanding the Concept of $x$-Intercept
The $x$-intercept of a graph is the point where the graph intersects the $x$-axis. In other words, it is the point where the value of $y$ is equal to zero. To find the $x$-intercept of a function, we need to set the function equal to zero and solve for $x$.
Setting the Function Equal to Zero
To find the $x$-intercept of the function $f(x)=x^2-16x+64$, we need to set the function equal to zero and solve for $x$. This can be done by using the quadratic formula or by factoring the quadratic expression.
Factoring the Quadratic Expression
The quadratic expression $x^2-16x+64$ can be factored as follows:
Setting the Factored Expression Equal to Zero
Now that we have factored the quadratic expression, we can set the factored expression equal to zero and solve for $x$:
Solving for $x$
To solve for $x$, we need to take the square root of both sides of the equation:
Finding the Value of $x$
Now that we have the equation $x-8 = 0$, we can solve for $x$ by adding 8 to both sides of the equation:
Conclusion
Therefore, the $x$-intercept of the graph of the function $f(x)=x^2-16x+64$ is $(8,0)$.
Understanding the Graph of the Function
The graph of the function $f(x)=x^2-16x+64$ is a parabola that opens upward. The vertex of the parabola is located at the point $(8,0)$, which is also the $x$-intercept of the graph.
Visualizing the Graph
To visualize the graph of the function, we can use a graphing calculator or a computer program to plot the function. The graph will show a parabola that opens upward, with the vertex located at the point $(8,0)$.
Conclusion
In conclusion, the $x$-intercept of the graph of the function $f(x)=x^2-16x+64$ is $(8,0)$. This can be found by setting the function equal to zero and solving for $x$.
Final Answer
The final answer is $(8,0)$.
What is the $x$-intercept of a function?
The $x$-intercept of a function is the point where the graph of the function intersects the $x$-axis. In other words, it is the point where the value of $y$ is equal to zero.
How do I find the $x$-intercept of a function?
To find the $x$-intercept of a function, you need to set the function equal to zero and solve for $x$. This can be done by using the quadratic formula or by factoring the quadratic expression.
What is the difference between the $x$-intercept and the $y$-intercept?
The $x$-intercept is the point where the graph of the function intersects the $x$-axis, while the $y$-intercept is the point where the graph of the function intersects the $y$-axis.
Can the $x$-intercept be a complex number?
Yes, the $x$-intercept can be a complex number. This occurs when the quadratic expression has no real solutions, but rather two complex solutions.
How do I determine if the $x$-intercept is real or complex?
To determine if the $x$-intercept is real or complex, you need to examine the discriminant of the quadratic expression. If the discriminant is positive, the $x$-intercept is real. If the discriminant is negative, the $x$-intercept is complex.
Can the $x$-intercept be a repeated root?
Yes, the $x$-intercept can be a repeated root. This occurs when the quadratic expression has a repeated factor, such as $(x-8)^2$.
How do I find the $x$-intercept of a function with a repeated root?
To find the $x$-intercept of a function with a repeated root, you need to set the function equal to zero and solve for $x$. This can be done by factoring the quadratic expression or by using the quadratic formula.
Can the $x$-intercept be a rational number?
Yes, the $x$-intercept can be a rational number. This occurs when the quadratic expression has a rational solution, such as $x = 8$.
How do I determine if the $x$-intercept is rational or irrational?
To determine if the $x$-intercept is rational or irrational, you need to examine the solutions to the quadratic equation. If the solutions are rational numbers, the $x$-intercept is rational. If the solutions are irrational numbers, the $x$-intercept is irrational.
Can the $x$-intercept be a negative number?
Yes, the $x$-intercept can be a negative number. This occurs when the quadratic expression has a negative solution, such as $x = -8$.
How do I find the $x$-intercept of a function with a negative solution?
To find the $x$-intercept of a function with a negative solution, you need to set the function equal to zero and solve for $x$. This can be done by factoring the quadratic expression or by using the quadratic formula.
Conclusion
In conclusion, the $x$-intercept of a function is the point where the graph of the function intersects the $x$-axis. To find the $x$-intercept, you need to set the function equal to zero and solve for $x$. The $x$-intercept can be a real or complex number, and it can be a rational or irrational number.
Final Answer
The final answer is $(8,0)$.