Problem 31 - Algebra and Trigonometry

Question

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift to the right \(\frac{1}{2}\) unit, shrink vertically by a factor of \(0.1,\) and shift downward 2 units

Step-by-Step Solution

Answer

The final transformed function is \( f(x) = 0.1 \left| x - \frac{1}{2} \right| - 2 \).

Step 1: Identify the base function

The base function given is \( f(x) = |x| \). This is an absolute value function with a V-shaped graph, where the vertex is at the origin (0,0).

Step 2: Apply the horizontal shift

To shift the graph of \( f(x) \) to the right by \( \frac{1}{2} \) unit, we replace \( x \) with \( x - \frac{1}{2} \). The modified function becomes \( f(x) = \left| x - \frac{1}{2} \right| \).

Step 3: Apply the vertical shrink

Shrink the graph vertically by a factor of \( 0.1 \). This is achieved by multiplying the entire function by \( 0.1 \). So, we have \( f(x) = 0.1 \left| x - \frac{1}{2} \right| \).

Step 4: Apply the downward shift

Shift the graph downward by 2 units. Subtract 2 from the transformed function in Step 3. Therefore, the final transformed function is \( f(x) = 0.1 \left| x - \frac{1}{2} \right| - 2 \).

Key Concepts

Absolute Value Function

The absolute value function is one of the most important functions in mathematics. It's symbolized as \( f(x) = |x| \). The graph of this function forms a distinctive V-shape that starts at the origin, point (0,0). The absolute function takes any real number, either positive or negative, and returns its non-negative value.
This means if you plug in \( x = -3 \), the output will be \( 3 \). Similarly, if \( x = 3 \), the output is still \( 3 \). In essence, the absolute value function "flips" any negative values to positive ones. This behavior is what creates the V-shaped graph, symmetrical about the vertical axis.

Horizontal Shift

Performing a horizontal shift involves moving the graph left or right along the x-axis. In our exercise, the function undergoes a right shift by \( \frac{1}{2} \) unit.
To achieve this, we modify the input of the function by subtracting \( \frac{1}{2} \) from \( x \). The equation then becomes \( f(x) = \left| x - \frac{1}{2} \right| \). This shift does not affect the shape of the graph but merely the position. The whole V-shaped graph is now centered at \( \frac{1}{2} \) instead of at 0.
This can be remembered as changing the points on the x-axis without altering the vertical distances from the axis.

Vertical Shrink

When a graph is vertically shrunk, its y-values are compressed towards the x-axis. This is accomplished by multiplying the function by a constant factor less than 1. In this exercise, a vertical shrink factor of 0.1 is applied, transforming the function to \( f(x) = 0.1\left| x - \frac{1}{2} \right| \).
This factor reduces the height of the graph, making it "shorter". Each y-value originally computed by the absolute value function is scaled down to only a tenth of its previous value.
While the x-position of the graph remains unchanged, the y-values draw closer to the axis, compressing the V-shape into a flatter form. Vertical shrinkage reduces the steepness of the graph, making the sides less inclined.

Downward Shift

A downward shift results in moving the entire graph down along the y-axis. In our example, the graph is shifted downward by 2 units. This transformation is achieved by subtracting \( 2 \) from the function.
Our final equation becomes \( f(x) = 0.1\left| x - \frac{1}{2} \right| - 2 \).
This operation affects only the vertical position of the graph. It implies every point on the graph is now 2 units lower than it was initially. Despite this vertical change, the horizontal attributes of the graph remain untouched.
The downward shift is a straightforward transformation, offering a neat way to position the graph lower without affecting its shape or width.