The Image Of A Parabolic Lens Is Traced Onto A Graph. The Function F ( X ) = 1 4 ( X + 8 ) ( X − 4 F(x)=\frac{1}{4}(x+8)(x-4 F ( X ) = 4 1 ​ ( X + 8 ) ( X − 4 ] Represents The Image. At Which Points Does The Image Cross The X X X -axis?A. ( − 8 , 0 (-8,0 ( − 8 , 0 ] And ( 4 , 0 (4,0 ( 4 , 0 ]B. ( 8 , 0 (8,0 ( 8 , 0 ] And

Introduction

In this article, we will explore the concept of a parabolic lens and its image on a graph. The function $f(x)=\frac{1}{4}(x+8)(x-4)$ represents the image of the parabolic lens. Our goal is to find the points at which the image crosses the x-axis.

Understanding the Function

The given function is a quadratic function in the form of $f(x)=a(x-h)(x-k)$, where $a$, $h$, and $k$ are constants. In this case, $a=\frac{1}{4}$, $h=-8$, and $k=4$. The graph of this function is a parabola that opens upwards, with its vertex at the point $(-8,0)$.

Finding the Points of Intersection with the x-Axis

To find the points at which the image crosses the x-axis, we need to find the values of $x$ for which $f(x)=0$. This means that we need to solve the equation $\frac{1}{4}(x+8)(x-4)=0$.

Solving the Equation

We can start by factoring the left-hand side of the equation:

$$\frac{1}{4}(x+8)(x-4)=0$$

$$\Rightarrow (x+8)(x-4)=0$$

Now, we can use the zero-product property to find the values of $x$ that satisfy the equation:

$$x+8=0 \quad \text{or} \quad x-4=0$$

Solving for $x$, we get:

$$x=-8 \quad \text{or} \quad x=4$$

Therefore, the points at which the image crosses the x-axis are $(-8,0)$ and $(4,0)$.

Conclusion

In this article, we have found the points at which the image of a parabolic lens crosses the x-axis. The function $f(x)=\frac{1}{4}(x+8)(x-4)$ represents the image of the parabolic lens, and we have solved the equation $\frac{1}{4}(x+8)(x-4)=0$ to find the values of $x$ for which $f(x)=0$. The points at which the image crosses the x-axis are $(-8,0)$ and $(4,0)$.

Answer

The correct answer is A. $(-8,0)$ and $(4,0)$.

Additional Information

• The parabolic lens is a type of optical lens that is used to focus light.
• The image of the parabolic lens is represented by the function $f(x)=\frac{1}{4}(x+8)(x-4)$.
• The points at which the image crosses the x-axis are $(-8,0)$ and $(4,0)$.

References

• [1] "Parabolic Lenses." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/technology/parabolic-lens.
• [2] "Quadratic Functions." Math Open Reference, mathopenref.com/quadratic.html.

Mathematical Concepts

• Quadratic functions
• Parabolic lenses
• Optical lenses
• Image formation

Key Terms

• Parabolic lens
• Quadratic function
• Image formation
• Optical lens

Mathematical Operations

• Factoring
• Zero-product property
• Solving equations

Mathematical Formulas

• $f(x)=\frac{1}{4}(x+8)(x-4)$
• $(x+8)(x-4)=0$
• $x+8=0 \quad \text{or} \quad x-4=0$

Mathematical Theorems

• Zero-product property
• Quadratic formula

Mathematical Proofs

• Proof of the zero-product property
• Proof of the quadratic formula
  The Image of a Parabolic Lens: Q&A =====================================

Q: What is a parabolic lens?

A: A parabolic lens is a type of optical lens that is used to focus light. It is a curved surface that is shaped like a parabola, and it is used to collect and focus light from a distant object.

Q: What is the function that represents the image of the parabolic lens?

A: The function that represents the image of the parabolic lens is $f(x)=\frac{1}{4}(x+8)(x-4)$.

Q: How do you find the points at which the image crosses the x-axis?

A: To find the points at which the image crosses the x-axis, you need to solve the equation $\frac{1}{4}(x+8)(x-4)=0$. This means that you need to find the values of $x$ for which $f(x)=0$.

Q: What are the values of $x$ that satisfy the equation $\frac{1}{4}(x+8)(x-4)=0$?

A: The values of $x$ that satisfy the equation $\frac{1}{4}(x+8)(x-4)=0$ are $x=-8$ and $x=4$.

Q: What are the points at which the image crosses the x-axis?

A: The points at which the image crosses the x-axis are $(-8,0)$ and $(4,0)$.

Q: What is the significance of the parabolic lens in optics?

A: The parabolic lens is significant in optics because it is used to focus light from a distant object. It is used in a variety of applications, including telescopes, microscopes, and cameras.

Q: What are some common applications of the parabolic lens?

A: Some common applications of the parabolic lens include:

• Telescopes: The parabolic lens is used in telescopes to collect and focus light from distant stars and galaxies.
• Microscopes: The parabolic lens is used in microscopes to collect and focus light from small objects.
• Cameras: The parabolic lens is used in cameras to collect and focus light from a scene.

Q: What are some common challenges associated with the parabolic lens?

A: Some common challenges associated with the parabolic lens include:

• Aberrations: The parabolic lens can suffer from aberrations, which are distortions in the image that can affect its quality.
• Diffraction: The parabolic lens can suffer from diffraction, which is the bending of light around the edges of the lens.
• Optical noise: The parabolic lens can suffer from optical noise, which is the random fluctuations in the light that can affect its quality.

Q: How can the challenges associated with the parabolic lens be overcome?

A: The challenges associated with the parabolic lens can be overcome by using advanced materials and manufacturing techniques to reduce aberrations and diffraction. Additionally, optical noise can be reduced by using noise-reducing techniques, such as image processing and filtering.

Q: What are some common materials used to make parabolic lenses?

A: Some common materials used to make parabolic lenses include:

• Glass: Glass is a common material used to make parabolic lenses because it is transparent, durable, and can be easily shaped.
• Plastic: Plastic is a common material used to make parabolic lenses because it is lightweight, flexible, and can be easily molded.
• Metal: Metal is a common material used to make parabolic lenses because it is durable, resistant to scratches, and can be easily polished.

Q: What are some common manufacturing techniques used to make parabolic lenses?

A: Some common manufacturing techniques used to make parabolic lenses include:

• Grinding: Grinding is a technique used to shape the lens by removing material from its surface.
• Polishing: Polishing is a technique used to smooth the surface of the lens and remove any imperfections.
• Coating: Coating is a technique used to apply a thin layer of material to the surface of the lens to reduce aberrations and improve its optical quality.

Q: What are some common applications of the parabolic lens in industry?

A: Some common applications of the parabolic lens in industry include:

• Surveillance: The parabolic lens is used in surveillance systems to collect and focus light from a scene.
• Inspection: The parabolic lens is used in inspection systems to collect and focus light from a surface.
• Measurement: The parabolic lens is used in measurement systems to collect and focus light from a surface.

Q: What are some common applications of the parabolic lens in medicine?

A: Some common applications of the parabolic lens in medicine include:

• Endoscopy: The parabolic lens is used in endoscopy to collect and focus light from the inside of the body.
• Microscopy: The parabolic lens is used in microscopy to collect and focus light from small objects.
• Imaging: The parabolic lens is used in imaging systems to collect and focus light from a scene.

Q: What are some common applications of the parabolic lens in astronomy?

A: Some common applications of the parabolic lens in astronomy include:

• Telescopes: The parabolic lens is used in telescopes to collect and focus light from distant stars and galaxies.
• Spectroscopy: The parabolic lens is used in spectroscopy to collect and focus light from a scene.
• Imaging: The parabolic lens is used in imaging systems to collect and focus light from a scene.

Q: What are some common applications of the parabolic lens in education?

A: Some common applications of the parabolic lens in education include:

• Optics: The parabolic lens is used in optics to teach students about the behavior of light.
• Physics: The parabolic lens is used in physics to teach students about the behavior of light and its interaction with matter.
• Engineering: The parabolic lens is used in engineering to teach students about the design and construction of optical systems.