If You Horizontally Stretch The Quadratic Parent Function, $f(x)=x^2$, By A Factor Of 3, What Is The Equation Of The New Function?A. $g(x)=\left(\frac{1}{3} X\right)^2$ B. $g(x)=(3 X)^2$ C. $g(x)=3 X^2$ D.

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Understanding the Quadratic Parent Function

The quadratic parent function, f(x)=x2f(x)=x^2, is a fundamental concept in algebra and mathematics. It represents a parabola that opens upwards, with its vertex at the origin (0,0). The function is symmetric about the y-axis and has a minimum point at the vertex.

Stretching the Function

When we stretch the quadratic parent function horizontally by a factor of 3, we are essentially compressing the function vertically by a factor of 3. This means that the new function will have a narrower and taller graph compared to the original function.

Determining the Equation of the New Function

To determine the equation of the new function, we need to consider the effect of the horizontal stretch on the original function. When we stretch the function horizontally by a factor of 3, the x-coordinates of the points on the graph are compressed by a factor of 3.

Applying the Horizontal Stretch

Let's consider a point (x, y) on the original graph of the quadratic parent function. When we stretch the function horizontally by a factor of 3, the new x-coordinate of the point will be 3x. The y-coordinate remains the same, as the vertical stretch does not affect the y-coordinates.

Deriving the Equation of the New Function

Using the new x-coordinate (3x), we can derive the equation of the new function. We know that the original function is f(x)=x2f(x)=x^2, so we can substitute 3x for x in the original equation:

g(x)=(3x)2g(x)=(3x)^2

Simplifying the Equation

We can simplify the equation by expanding the squared term:

g(x)=9x2g(x)=9x^2

Comparing with the Options

Now, let's compare the derived equation with the options provided:

  • A. g(x)=(13x)2g(x)=\left(\frac{1}{3} x\right)^2
  • B. g(x)=(3x)2g(x)=(3 x)^2
  • C. g(x)=3x2g(x)=3 x^2
  • D. (no option)

The correct equation is option B, g(x)=(3x)2g(x)=(3 x)^2. This equation represents the quadratic parent function stretched horizontally by a factor of 3.

Conclusion

In conclusion, when we stretch the quadratic parent function horizontally by a factor of 3, the equation of the new function is g(x)=(3x)2g(x)=(3 x)^2. This equation represents a parabola that opens upwards, with its vertex at the origin (0,0), but with a narrower and taller graph compared to the original function.

Key Takeaways

  • The quadratic parent function, f(x)=x2f(x)=x^2, is a fundamental concept in algebra and mathematics.
  • Stretching the function horizontally by a factor of 3 compresses the function vertically by a factor of 3.
  • The equation of the new function is g(x)=(3x)2g(x)=(3 x)^2.
  • The new function has a narrower and taller graph compared to the original function.
    Quadratic Function Stretching: Q&A =====================================

Frequently Asked Questions

Q: What is the effect of stretching the quadratic parent function horizontally by a factor of 3?

A: When we stretch the quadratic parent function horizontally by a factor of 3, we are essentially compressing the function vertically by a factor of 3. This means that the new function will have a narrower and taller graph compared to the original function.

Q: How do I determine the equation of the new function after stretching the quadratic parent function horizontally?

A: To determine the equation of the new function, we need to consider the effect of the horizontal stretch on the original function. We can derive the equation of the new function by substituting 3x for x in the original equation.

Q: What is the equation of the new function after stretching the quadratic parent function horizontally by a factor of 3?

A: The equation of the new function is g(x)=(3x)2g(x)=(3 x)^2. This equation represents a parabola that opens upwards, with its vertex at the origin (0,0), but with a narrower and taller graph compared to the original function.

Q: How does the horizontal stretch affect the graph of the quadratic parent function?

A: The horizontal stretch compresses the function vertically by a factor of 3, resulting in a narrower and taller graph compared to the original function.

Q: Can I apply the horizontal stretch to other types of functions?

A: Yes, the horizontal stretch can be applied to other types of functions, not just the quadratic parent function. However, the effect of the horizontal stretch will depend on the specific function being stretched.

Q: What is the relationship between the horizontal stretch and the vertical stretch?

A: The horizontal stretch and the vertical stretch are related in that they are inverse operations. When we stretch a function horizontally by a factor of 3, we are compressing the function vertically by a factor of 3.

Q: How do I apply the horizontal stretch to a function in a specific problem?

A: To apply the horizontal stretch to a function in a specific problem, you need to identify the function being stretched and determine the factor by which it is being stretched. Then, you can substitute the new x-coordinate into the original equation to derive the equation of the new function.

Q: What are some common applications of the horizontal stretch in mathematics?

A: The horizontal stretch has many applications in mathematics, including:

  • Graphing functions
  • Analyzing the behavior of functions
  • Solving equations and inequalities
  • Modeling real-world phenomena

Q: Can I use the horizontal stretch to solve problems in other fields, such as physics or engineering?

A: Yes, the horizontal stretch can be used to solve problems in other fields, such as physics or engineering. The horizontal stretch is a fundamental concept in mathematics that has many applications in other fields.

Conclusion

In conclusion, the horizontal stretch is a powerful tool in mathematics that can be used to analyze and solve problems involving functions. By understanding the effect of the horizontal stretch on the quadratic parent function, we can apply this concept to other types of functions and solve a wide range of problems in mathematics and other fields.

Key Takeaways

  • The horizontal stretch compresses the function vertically by a factor of 3.
  • The equation of the new function is g(x)=(3x)2g(x)=(3 x)^2.
  • The horizontal stretch has many applications in mathematics and other fields.
  • The horizontal stretch is a fundamental concept in mathematics that can be used to solve a wide range of problems.