Which Function Has Only One $x$-intercept At $(-6,0)$?A. $f(x)=x(x-6)$ B. $f(x)=(x-6)(x-6)$ C. $f(x)=(x+6)(x-6)$ D. $f(x)=(x+6)(x+6)$

Understanding $x$-Intercepts

In mathematics, an $x$-intercept is a point where a graph intersects the $x$-axis. This occurs when the $y$-coordinate of the point is equal to zero. In other words, if we have a function $f(x)$, the $x$-intercept is the point where $f(x) = 0$. Understanding $x$-intercepts is crucial in various mathematical concepts, including algebra, geometry, and calculus.

Analyzing the Given Functions

To determine which function has only one $x$-intercept at $(-6,0)$, we need to analyze each of the given functions:

Function A: $f(x) = x(x - 6)$

Let's expand the function:

$$f(x) = x^2 - 6x$$

To find the $x$-intercept, we set $f(x) = 0$:

$$x^2 - 6x = 0$$

Factoring out $x$, we get:

$$x(x - 6) = 0$$

This gives us two possible values for $x$: $x = 0$ and $x = 6$. Therefore, function A has two $x$-intercepts: $(0,0)$ and $(6,0)$.

Function B: $f(x) = (x - 6)(x - 6)$

Expanding the function, we get:

$$f(x) = x^2 - 12x + 36$$

To find the $x$-intercept, we set $f(x) = 0$:

$$x^2 - 12x + 36 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -12$, and $c = 36$. Plugging these values into the formula, we get:

$$x = \frac{12 \pm \sqrt{(-12)^2 - 4(1)(36)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{12 \pm \sqrt{144 - 144}}{2}$$

$$x = \frac{12 \pm \sqrt{0}}{2}$$

$$x = \frac{12}{2}$$

$$x = 6$$

Therefore, function B has only one $x$-intercept at $(6,0)$.

Function C: $f(x) = (x + 6)(x - 6)$

Expanding the function, we get:

$$f(x) = x^2 - 36$$

To find the $x$-intercept, we set $f(x) = 0$:

$$x^2 - 36 = 0$$

This is a quadratic equation, and we can solve it by adding 36 to both sides:

$$x^2 = 36$$

Taking the square root of both sides, we get:

$$x = \pm \sqrt{36}$$

$$x = \pm 6$$

Therefore, function C has two $x$-intercepts: $(6,0)$ and $(-6,0)$.

Function D: $f(x) = (x + 6)(x + 6)$

Expanding the function, we get:

$$f(x) = x^2 + 12x + 36$$

To find the $x$-intercept, we set $f(x) = 0$:

$$x^2 + 12x + 36 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = 12$, and $c = 36$. Plugging these values into the formula, we get:

$$x = \frac{-12 \pm \sqrt{(12)^2 - 4(1)(36)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-12 \pm \sqrt{144 - 144}}{2}$$

$$x = \frac{-12 \pm \sqrt{0}}{2}$$

$$x = \frac{-12}{2}$$

$$x = -6$$

Therefore, function D has only one $x$-intercept at $(-6,0)$.

Conclusion

Based on our analysis, we can conclude that function B and function D have only one $x$-intercept at $(6,0)$ and $(-6,0)$, respectively. However, since the question asks for a function with an $x$-intercept at $(-6,0)$, the correct answer is function D: $f(x) = (x + 6)(x + 6)$.

Understanding $x$-Intercepts: A Comprehensive Guide

In our previous article, we explored the concept of $x$-intercepts and analyzed four different functions to determine which one has only one $x$-intercept at $(-6,0)$. In this article, we will answer some frequently asked questions about $x$-intercepts and provide a comprehensive guide to help you understand this important mathematical concept.

Q: What is an $x$-intercept?

A: An $x$-intercept is a point where a graph intersects the $x$-axis. This occurs when the $y$-coordinate of the point is equal to zero. In other words, if we have a function $f(x)$, the $x$-intercept is the point where $f(x) = 0$.

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

A: 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 using various methods, including factoring, the quadratic formula, or graphing.

Q: What is the difference between an $x$-intercept and a $y$-intercept?

A: An $x$-intercept is a point where a graph intersects the $x$-axis, while a $y$-intercept is a point where a graph intersects the $y$-axis. In other words, an $x$-intercept occurs when $y = 0$, while a $y$-intercept occurs when $x = 0$.

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

A: Yes, a function can have more than one $x$-intercept. In fact, most functions have multiple $x$-intercepts, which can be found by setting the function equal to zero and solving for $x$.

Q: How do I determine the number of $x$-intercepts a function has?

A: To determine the number of $x$-intercepts a function has, you need to analyze the function's equation and solve for $x$. If the equation has multiple solutions, the function has multiple $x$-intercepts.

Q: Can a function have no $x$-intercepts?

A: Yes, a function can have no $x$-intercepts. This occurs when the function's equation has no real solutions, meaning that there is no point where the graph intersects the $x$-axis.

Q: What is the significance of $x$-intercepts in real-world applications?

A: $x$-intercepts have significant importance in various real-world applications, including physics, engineering, and economics. For example, in physics, the $x$-intercept of a graph can represent the point where a particle or object intersects the $x$-axis, while in economics, the $x$-intercept of a graph can represent the point where a company's revenue or cost intersects the $x$-axis.

Conclusion

In conclusion, $x$-intercepts are an essential concept in mathematics, and understanding them is crucial for various real-world applications. By analyzing the function's equation and solving for $x$, you can determine the number of $x$-intercepts a function has and understand the significance of $x$-intercepts in real-world applications.

Additional Resources

For further learning, we recommend the following resources:

• Khan Academy: $x$-Intercepts
• Mathway: $x$-Intercepts
• Wolfram Alpha: $x$-Intercepts

By exploring these resources, you can gain a deeper understanding of $x$-intercepts and improve your math skills.