Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=|2 x|+4\); horizontal shrink by a factor of \(\frac{1}{2}\)

Short answer

The transformed function is \(g(x) = |4x| + 4\)

Step-by-step solution

Understand the transformation

A horizontal shrink by a factor of 1/2 of the function \(f(x) = |2x|+4\) means that we multiply the independent variable, x, by the reciprocal of 1/2, which is 2.

Applying the transformation to the function

Apply the transformation to the function \(f(x) = |2x| + 4\), replacing x with 2x. The transformed function becomes \(g(x) = |2*(2x)| +4 = |4x| + 4.\)

Checking the solution

Check the solution using a graphing calculator. Plot both \(f(x) = |2x| + 4\) and \(g(x) = |4x| +4\) and observe that the graph of g(x) is a horizontally shrunken version of the graph of f(x).\)

Key concepts

Horizontal Shrink

When discussing function transformation, one particular type is a horizontal shrink. This occurs when each point on the graph of a function moves closer to the y-axis, compressing the graph horizontally. It’s like squeezing a rubber band from the sides; the band gets narrower but remains in the same position on the floor.

In mathematical terms, a horizontal shrink by a factor of 1/2, as in our exercise with the function f(x) = |2x|+4, means we scale the x-values by a factor greater than 1. In this case, every x in our original function is replaced with 2x, effectively halving the distance from each x-value to the y-axis. The general formula for a horizontal shrink is to replace x with x/c, where c is the shrink factor.

Absolute Value Function

An absolute value function is represented by the formula f(x) = |x|, where |x| denotes the absolute value of x. The absolute value is a measure of distance that's always nonnegative, showing how far x is from zero on a number line, without regard to direction.

The graph of an absolute value function is a V-shaped curve that opens upwards. The point at the V, called the vertex, occurs at the origin for the basic function f(x) = |x|. When we introduce coefficients, like f(x) = |2x|, the slope of the arms of the V changes, making the graph steeper or flatter.

Graphing Calculator

A graphing calculator is an indispensable tool for visualizing function transformations. It can plot the graphs of different functions, allowing students to compare the forms and understand the effects of transformations like stretching or shrinking.

To verify the horizontal shrink of our function f(x) using a graphing calculator, you would enter both the original function and the transformed function. By viewing them together on the screen, you can clearly see how one graph is a compressed version of the other along the horizontal axis. This visual representation aids in solidifying your understanding of the transformation applied.

Function Transformation

A function transformation takes a basic function and applies a set of changes to its graph. These can include shifts, reflections, stretches, and shrinks. When we talk about horizontally shrinking a function like f(x) = |2x|+4 by a factor of 1/2, we’re applying a horizontal scaling transformation to it.

Understanding how to manipulate the function's equation to achieve the desired transformation is crucial. In our example, we multiply the input by 2 to achieve a horizontal shrink by a factor of 1/2, resulting in the function g(x) = |4x|+4. Checking this transformation with a graphing calculator not only confirms the theory but also provides a visual aid, making the concept easier to comprehend.