Math Problem Statement

Suppose f(x) = √x. Provide the function for each of the following transformations: reflection across the x-axis, left shift, down shift, right shift and up shift, and vertical stretch.

Solution

Let's break down and answer each part of the problem from the images you've uploaded.

Original Function:

f(x)=xf(x) = \sqrt{x}

1. Reflection across the x-axis:

When reflecting a function across the x-axis, we multiply the entire function by 1-1. This gives: f(x)=xf(x) = -\sqrt{x}

So, the correct choice is: f(x)=xf(x) = -\sqrt{x}

2. Shift left by 2 units:

To shift a function left by kk units, replace xx with x+kx + k. In this case, shifting left by 2 units means replacing xx with x+2x + 2, giving: f(x)=x+2f(x) = \sqrt{x + 2}

So, the correct choice is: f(x)=x+2f(x) = \sqrt{x + 2}

3. Shift down by 1 unit:

To shift a function down by kk units, subtract kk from the entire function. For a downward shift of 1 unit: f(x)=x1f(x) = \sqrt{x} - 1

So, the correct choice is: f(x)=x1f(x) = \sqrt{x} - 1

4. Shift right by 2 units and up by 1 unit:

To shift a function right by kk units, replace xx with xkx - k, and to shift up by mm units, add mm to the entire function. So, shifting right by 2 and up by 1 gives: f(x)=x2+1f(x) = \sqrt{x - 2} + 1

So, the correct choice is: f(x)=x2+1f(x) = \sqrt{x - 2} + 1

5. Vertical stretch by a factor of 3:

To stretch a function vertically by a factor of aa, multiply the entire function by aa. For a vertical stretch by 3: f(x)=3xf(x) = 3\sqrt{x}

So, the correct choice is: f(x)=3xf(x) = 3\sqrt{x}


Would you like further details on these transformations or any other questions about the problems? Here are some additional practice questions you can consider:

  1. How do you reflect a function across the y-axis?
  2. What is the effect of a horizontal stretch on f(x)f(x)?
  3. How would you describe the transformation f(x)=2x3+4f(x) = -2\sqrt{x - 3} + 4?
  4. What happens to the domain and range after a vertical stretch of f(x)f(x)?
  5. How do transformations affect the intercepts of a function?

Tip: Always apply horizontal transformations (like shifts or stretches) inside the function's argument and vertical ones outside.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Function Transformations
Square Root Functions
Algebra

Formulas

Reflection across x-axis: f(x) = -f(x)
Horizontal shift: f(x - h) (right) or f(x + h) (left)
Vertical shift: f(x) + k (up) or f(x) - k (down)
Vertical stretch: a * f(x), where a > 1 for stretching

Theorems

-

Suitable Grade Level

Grades 9-12