Describe The Effect { F\left(\frac{1}{4} X\right) $}$ Will Have On The Function { F(x) = X^2 $}$.A. The Vertical Coordinates Will Increase Their Distance From The { X $}$-axis By A Factor Of 4.B. The Vertical

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Introduction

In mathematics, functions are used to describe the relationship between variables. When a function is transformed, its graph changes in a predictable way. In this article, we will explore the effect of a horizontal stretch on a quadratic function. Specifically, we will examine how the function { f(x) = x^2 $}$ changes when its input is multiplied by { \frac{1}{4} $}$.

The Original Function

The original function is a quadratic function given by { f(x) = x^2 $}$. This function has a parabolic shape, with its vertex at the origin (0, 0). The graph of this function is a U-shaped curve that opens upwards.

The Horizontal Stretch

When the input of the function is multiplied by { \frac{1}{4} $}$, the function becomes { f\left(\frac{1}{4} x\right) $}$. This is an example of a horizontal stretch, where the input is scaled by a factor of { \frac{1}{4} $}$. To understand the effect of this transformation, let's consider what happens to the graph of the function.

Effect on the Graph

When the input is multiplied by { \frac{1}{4} $}$, the graph of the function is stretched horizontally by a factor of 4. This means that the x-coordinates of the points on the graph are multiplied by { \frac{1}{4} $}$. As a result, the vertical coordinates of the points on the graph are also affected.

Vertical Coordinates

To understand how the vertical coordinates are affected, let's consider a point on the graph of the original function. Suppose the point has coordinates (x, y). When the input is multiplied by { \frac{1}{4} $}$, the new x-coordinate is { \frac{1}{4} x $}$. The corresponding y-coordinate is given by the function { f\left(\frac{1}{4} x\right) $}$.

Analyzing the Transformation

To analyze the transformation, let's substitute { \frac{1}{4} x $}$ into the original function { f(x) = x^2 $}$. This gives us:

{ f\left(\frac{1}{4} x\right) = \left(\frac{1}{4} x\right)^2 $}$

Simplifying this expression, we get:

{ f\left(\frac{1}{4} x\right) = \frac{1}{16} x^2 $}$

This shows that the vertical coordinates of the points on the graph are multiplied by { \frac{1}{16} $}$. Since { \frac{1}{16} $}$ is equal to { \left(\frac{1}{4}\right)^2 $}$, we can conclude that the vertical coordinates are increased by a factor of 4.

Conclusion

In conclusion, when the input of the function { f(x) = x^2 $}$ is multiplied by { \frac{1}{4} $}$, the function becomes { f\left(\frac{1}{4} x\right) $}$. This is an example of a horizontal stretch, where the input is scaled by a factor of { \frac{1}{4} $}$. As a result, the vertical coordinates of the points on the graph are increased by a factor of 4.

Key Takeaways

  • When the input of a function is multiplied by a factor, the graph of the function is stretched horizontally by the reciprocal of that factor.
  • The vertical coordinates of the points on the graph are affected by the horizontal stretch.
  • In the case of a quadratic function, the vertical coordinates are increased by a factor equal to the square of the reciprocal of the horizontal stretch factor.

Example Problems

  1. Suppose the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 2. What is the new function?
  2. If the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 3, what is the new function?
  3. When the input of the function { f(x) = x^2 $}$ is multiplied by { \frac{1}{2} $}$, what is the new function?

Solutions

  1. The new function is { f\left(\frac{1}{2} x\right) = \left(\frac{1}{2} x\right)^2 $}$.
  2. The new function is { f\left(\frac{1}{3} x\right) = \left(\frac{1}{3} x\right)^2 $}$.
  3. The new function is { f\left(\frac{1}{2} x\right) = \frac{1}{4} x^2 $}$.

Practice Problems

  1. Suppose the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 5. What is the new function?
  2. If the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 6, what is the new function?
  3. When the input of the function { f(x) = x^2 $}$ is multiplied by { \frac{1}{3} $}$, what is the new function?

Solutions

  1. The new function is { f\left(\frac{1}{5} x\right) = \left(\frac{1}{5} x\right)^2 $}$.
  2. The new function is { f\left(\frac{1}{6} x\right) = \left(\frac{1}{6} x\right)^2 $}$.
  3. The new function is { f\left(\frac{1}{3} x\right) = \frac{1}{9} x^2 $}$.

Conclusion

Frequently Asked Questions

Q: What is a horizontal stretch in a function?

A: A horizontal stretch in a function is a transformation where the input of the function is multiplied by a factor. This causes the graph of the function to stretch horizontally by the reciprocal of that factor.

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

A: When a quadratic function is stretched horizontally, the graph of the function is stretched horizontally by the reciprocal of the stretch factor. The vertical coordinates of the points on the graph are affected by the horizontal stretch.

Q: What is the effect of a horizontal stretch on the vertical coordinates of a quadratic function?

A: In the case of a quadratic function, the vertical coordinates are increased by a factor equal to the square of the reciprocal of the horizontal stretch factor.

Q: Can you provide an example of a horizontal stretch on a quadratic function?

A: Suppose the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 2. The new function is { f\left(\frac{1}{2} x\right) = \left(\frac{1}{2} x\right)^2 $}$.

Q: How do you determine the new function after a horizontal stretch?

A: To determine the new function after a horizontal stretch, you multiply the input of the original function by the reciprocal of the stretch factor.

Q: What is the reciprocal of a stretch factor?

A: The reciprocal of a stretch factor is 1 divided by the stretch factor. For example, the reciprocal of a stretch factor of 2 is { \frac{1}{2} $}$.

Q: Can you provide another example of a horizontal stretch on a quadratic function?

A: Suppose the function { f(x) = x^2 $}$ is stretched horizontally by a factor of 3. The new function is { f\left(\frac{1}{3} x\right) = \left(\frac{1}{3} x\right)^2 $}$.

Q: How do you determine the effect of a horizontal stretch on the graph of a function?

A: To determine the effect of a horizontal stretch on the graph of a function, you need to consider the reciprocal of the stretch factor and the square of that reciprocal.

Q: Can you provide a summary of the key points about horizontal stretches on quadratic functions?

A: Here is a summary of the key points:

  • A horizontal stretch in a function is a transformation where the input of the function is multiplied by a factor.
  • The graph of a quadratic function is stretched horizontally by the reciprocal of the stretch factor.
  • The vertical coordinates of the points on the graph are affected by the horizontal stretch.
  • In the case of a quadratic function, the vertical coordinates are increased by a factor equal to the square of the reciprocal of the horizontal stretch factor.

Common Mistakes to Avoid

  • Confusing the stretch factor with the reciprocal of the stretch factor.
  • Failing to consider the square of the reciprocal of the stretch factor when determining the effect of a horizontal stretch on the vertical coordinates.
  • Not understanding the relationship between the horizontal stretch factor and the reciprocal of that factor.

Conclusion

In conclusion, a horizontal stretch in a function is a transformation where the input of the function is multiplied by a factor. The graph of a quadratic function is stretched horizontally by the reciprocal of the stretch factor, and the vertical coordinates of the points on the graph are affected by the horizontal stretch. By understanding the effect of a horizontal stretch on a quadratic function, you can better analyze and solve problems involving function transformations.