Question

Thomas and Sharyn discuss the order of the transformations of the graph of \(y=-3|x|\) compared to the graph of \(y=|x|\) Thomas states that the reflection must be applied first. Sharyn claims that the vertical stretch should be applied first. a) Sketch the graph of \(y=-3|x|\) by applying the reflection first. b) Sketch the graph of \(y=-3|x|\) by applying the stretch first. c) Explain your conclusions. Who is correct?

Short answer

Both Thomas and Sharyn are correct. Reflections or stretches can be applied in either order, and the graph of \( y = -3|x| \) remains the same.

Step-by-step solution

Understand the Basic Graph

The graph of the function \( y = |x| \) is a V-shaped graph with its vertex at the origin (0,0). The arms of the graph extend upwards with a slope of 1 and -1.

Apply the Reflection First (Thomas's View)

To apply the reflection of \( y = |x| \) in the x-axis, change the sign of the entire function. So, the reflection of \( y = |x| \) is \( y = -|x| \). This graph is still V-shaped but opens downward instead of upward.

Apply the Vertical Stretch Second (Thomas's View)

Next, apply a vertical stretch by a factor of 3 to the function \( y = -|x| \). Multiply the output by 3 to obtain the function \( y = -3|x| \). This changes the slope of the arms from -1 to -3 and the graph remains opening downward.

Sketch the Graph (Thomas's View)

Draw the V-shaped graph opening downwards with its vertex at (0,0). The slopes of the two lines are now -3 and -3 respectively.

Apply the Vertical Stretch First (Sharyn's View)

To apply the vertical stretch of \( y = |x| \) by a factor of 3, multiply the output by 3. So, \( y = |x| \) becomes \( y = 3|x| \). This graph is V-shaped and opens upward but the slope of the arms is now 3 and -3.

Apply the Reflection Second (Sharyn's View)

Next, apply the reflection to \( y = 3|x| \) by changing the sign of the output. This changes the function to \( y = -3|x| \). The graph will now be V-shaped opening downward with slopes of -3 and -3.

Sketch the Graph (Sharyn's View)

Draw the V-shaped graph opening downwards with its vertex at (0,0). The slopes of the two lines should both be -3, just like in Step 4.

Conclusion

Both graphs, obtained by following Thomas's order (reflection first) and Sharyn's order (stretch first), are identical. Therefore, both Thomas and Sharyn are correct, as the final graph \( y = -3|x| \) is the same regardless of whether the reflection or stretch is applied first.

Key concepts

Reflection

In function transformations, a **reflection** flips the graph of a function over a specific axis. For this exercise, the reflection is over the x-axis. Consider the function: $$ y = |x| $$ This is a V-shaped graph, opening upwards. When we apply a reflection over the x-axis, the signs of the y-values are changed. So, the reflected function becomes: $$ y = -|x| $$ The graph now opens downwards, still maintaining its V-shape, but with the arms pointing down instead of up.

Vertical Stretch

A **vertical stretch** scales the graph of a function vertically by a certain factor. This factor is multiplied to all the y-values of the function. Starting with: $$ y = |x| $$ If we apply a vertical stretch by a factor of 3, each y-value on the graph is multiplied by 3, resulting in: $$ y = 3|x| $$ The V-shaped graph continues to open upwards, but the arms become steeper, multiplying the slopes. So, instead of the original slopes of 1 and -1, they become 3 and -3.

Absolute Value Functions

An **absolute value function** is defined as: $$ y = |x| $$ This function produces a V-shaped graph centered at the origin (0,0). The absolute value function ensures that all outputs are non-negative, thus graphing only in the first and second quadrants initially. For any transformations such as reflections or vertical stretches, it's essential to start with understanding this basic shape. Transformations will only modify the shape, direction, or scale of this graph.

Graph Sketching

When **sketching a graph** after transformations, follow these steps: - **Identify the base function**: Start with the most basic form, like \(y = |x|\). - **Apply transformations step-by-step**: Reflect over x-axis, apply vertical stretches or shrinks, etc. - **Draw transformed graph**: Take note of changes in vertex, slope, and direction. In this exercise, the base function \( y = |x| \) was transformed by reflecting it and applying a vertical stretch by a factor of 3, resulting in \( y = -3|x| \). Whether reflection or stretch is applied first doesn't change the final graph. This means consistency in method ensures correct sketch.«