Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=-2|x-4|+2\)

Short answer

The function \(g(x)\) that represents the transformation of \(f(x)\) is \(g(x)= -2| x-4| +2\).

Step-by-step solution

Identify the Base Function

The base function in this case is \(f(x) = |x|\). An absolute function always returns the positive value of its argument, giving a 'V' shape on the graph. Hence, the base function is the unmodified form of the variable. Here, the base function is \(|x|\).

Identify the Transformations

The transformed function is given by \(f(x) = -2|x-4|+2\). The transformations are described by the constants in this function. The '-2' in front of the absolute value indicates a vertical flip and vertical stretch by a factor of 2. The '-4' inside the absolute value function shifts the function 4 units to the right. The '+2' at the end of the function shifts the graph vertically upwards by 2 units.

Combine the Transformations to Form the New function g(x)

Now that the alterations have been determined, replace \(f(x)\) with \(g(x)\) to indicate the transformation of \(f(x)\). The new function \(g(x)\) that represents the transformation of \(f(x)\) will contain all identified transformations. So, \(g(x)= -2| x-4 |+2\).

Check the solution with a graphing calculator

Verify the proposed solution using a graphing calculator. Plot \(f(x)=|x|\) and then plot \(g(x)=-2| x-4 |+2\) on the same graph. Observe that \(g(x)\) is indeed a transformed version of \(f(x)\).

Key concepts

Absolute Value Functions

Absolute value functions are a type of piecewise function that always provides a non-negative output, regardless of the input. In mathematical terms, if our base function is given by \( f(x) = |x| \), the output is simply the absolute value of \( x \).
This results in a distinct 'V' shape graph that opens upwards, centered at the origin (0,0). This 'V' shape is due to the way absolute values convert negative inputs into positive outputs and leave positive inputs unchanged.

• The sharp corner or vertex of the 'V' occurs where \( x = 0 \), marking the minimum point of the function.
• The graph is symmetric about the y-axis, showing that \( f(x) = f(-x) \).

Understanding this base behavior is crucial, as transformations often modify this shape or its position on the graph. Such transformations are what allow functions to model a wider variety of real-world phenomena.

Vertical and Horizontal Shifts

In mathematics, a shift refers to moving a graph up, down, left, or right. These can be either vertical or horizontal shifts, modifying the function's graph position without altering its basic shape.
For vertical shifts, consider the function \( f(x) = |x| + k \). Here, \( k \) moves the graph up or down:

• If \( k > 0 \): the graph shifts upward by \( k \) units.
• If \( k < 0 \): the graph shifts downward by \( |k| \) units.

Horizontal shifts involve altering the input of the function, for example \( f(x) = |x - h| \). This affects its position along the x-axis:

• If \( h > 0 \): the graph shifts right by \( h \) units.
• If \( h < 0 \): the graph shifts left by \( |h| \) units.

These shifts allow adjustments to the function's intercepts, enabling better alignment with desired data or applications.

Vertical Stretching and Flipping

Vertical stretching and flipping are transformations that change the orientation and the steepness of the graph of a function. This is achieved by multiplying the function by a constant factor.
The expression \( f(x) = a|x| \) helps illustrate these transformations:

• If \( a > 1 \): the graph is vertically stretched, making it narrower and steeper.
• If \( 0 < a < 1 \): the graph is vertically compressed, making it wider and less steep.
• If \( a < 0 \): the graph is flipped over the x-axis, inverting the 'V' shape.

In the example \( f(x) = -2|x-4|+2 \), the factor of \(-2\) not only vertically stretches the graph by a factor of 2, but also flips it upside down. This causes the arms of the 'V' to open downward, rather than upward. Understanding these transformations helps in accurately sketching and interpreting graphs, essential skills in algebra.