Question

Gil is asked to translate the graph of \(y=|x|\) according to the equation \(y=|2 x-6|+2\) He decides to do the horizontal translation of 3 units to the right first, then the stretch about the \(y\) -axis by a factor of \(\frac{1}{2},\) and lastly the translation of 2 units up. This gives him Graph 1. To check his work, he decides to apply the horizontal stretch about the \(y\) -axis by a factor of \(\frac{1}{2}\) first, and then the horizontal translation of 6 units to the right and the vertical translation of 2 units up. This results in Graph 2. a) Explain why the two graphs are in different locations. b) How could Gil have rewritten the equation so that the order in which he did the transformations for Graph 2 resulted in the same position as Graph \(1 ?\)

Short answer

The graphs are in different locations because the order of transformations affects the intermediate steps. Rewriting the equation for Graph 2 as \( y = \left| \frac{x}{2} -3 \right| + 2 \) aligns it with Graph 1.

Step-by-step solution

Original Graph

Start with the basic graph of the absolute value function: \[ y = |x| \]

Horizontal Translation

To perform a horizontal translation of 3 units to the right, modify the equation:\[ y = |x-3| \]

Horizontal Stretch

Next, apply the stretch about the y-axis by a factor of \( \frac{1}{2} \). Here's how the new equation looks:\[ y = \left| \frac{1}{2} (x-3) \right| = \left| \frac{x-3}{2} \right| \]

Vertical Translation

Finally, perform the vertical translation of 2 units up. The final equation for Graph 1 is:\[ y = \left| \frac{ x-3}{2} \right| + 2 \]

Horizontal Stretch First

Now, let's follow the order given for Graph 2. Start with the horizontal stretch about the y-axis by a factor of \( \frac{1}{2} \):\[ y = | \frac{1}{2} x| \]

Horizontal Translation

Next, apply the horizontal translation of 6 units to the right:\[ y = \left| \frac{1}{2}(x-6) \right| = \left| \frac{x-6}{2} \right| \]

Vertical Translation

Finally, add the vertical translation of 2 units up:\[ y = \left| \frac{x-6}{2} \right| + 2 \]

Compare Graph 1 and Graph 2

Graph 1 equation: \[ y = \left| \frac{ x-3}{2} \right| + 2 \] Graph 2 equation: \[ y = \left| \frac{x-6}{2} \right| + 2 \] These two graphs are not the same because the horizontal translations were applied differently in each sequence, resulting in different intermediate points before the final transformation.

Rewriting the Equation

To rewrite the equation for Graph 2's transformations to result in the same position as Graph 1, start with the original function and apply the transformations in this order: - Factor the inside as \(\left| \frac{1}{2} (x-6) \right| \)- Adjust the right translation to 3 units (considering the stretch): \[ y = \left| \frac{x}{2} -3 \right| + 2 \]

Key concepts

absolute value function

The absolute value function, denoted as \(y = |x|\), is a function that takes any real number x and returns its absolute value. The graph of \(y = |x|\) is V-shaped and is symmetric with respect to the y-axis. The 'V' opens upwards, and its vertex is located at the origin (0,0). This function is important in real-life contexts where only the magnitude matters, without considering the direction.
For example:

• \(|-3| = 3\)
• \(|4| = 4\)

Understanding the basic shape and properties of the absolute value function is key before moving on to graph transformations.

horizontal translation

A horizontal translation shifts the graph left or right along the x-axis. This shift is represented inside the function's parentheses. For example, to move the graph of \(y = |x|\) three units to the right, you would modify the equation to \(y = |x-3|\). This transformation means every point on the graph moves to the right by 3 units.
Key Points:

• If the translation is to the right: \[ y = |x - a| \]
• If the translation is to the left: \[ y = |x + a| \]

The direction depends on the sign of the constant inside the function. Always remember: negative moves right, positive moves left.

vertical translation

A vertical translation shifts the graph up or down along the y-axis. This shift is represented outside the absolute value function. For instance, to move the graph of \(y = |x|\) two units up, you would modify the equation to \(y = |x| + 2\). This transformation means every point on the graph moves up by 2 units.
Key Points:

• If the translation is up: \[ y = |x| + b \]
• If the translation is down: \[ y = |x| - b \]

The direction depends on the sign of the constant added or subtracted outside the function. Positive values move the graph up, while negative values move it down.

horizontal stretch

A horizontal stretch changes the width of the graph. This transformation is applied inside the function by multiplying the x variable by a factor. For example, to horizontally stretch the graph of \(y = |x|\) by a factor of \(\frac{1}{2}\), you modify the equation to \(y = \left| \frac{1}{2}x \right|\). This transformation compresses the graph so that each x-coordinate is halved.
Key Points:

• A factor greater than 1 compresses the graph.
• A factor between 0 and 1 stretches the graph.
• The usual form is: \[ y = \left| \frac{x}{k} \right| \]

Understanding how the graph is altered visually by multiplication helps in interpreting and predicting the effects of transformations.