Problem 30 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=-\frac{2}{5}|-x|+6\) can be obtained from the graph of \(y=|x|\) by reflecting across the __________ -axis, vertically shrinking by applying a factor of __________ reflecting across the _________ -axis, and shifting vertically _________ units in the _______ direction.

Step-by-Step Solution

Answer

x, 2/5, x, 6, up

Step 1: Identify the Base Function

The base function given is \( y = |x| \). This is a V-shaped graph that opens upwards with the vertex at the origin (0, 0).

Step 2: Reflect Across the Y-Axis

The expression \(-|-x|\) indicates a reflection across the y-axis. However, \(|-x|\) is equivalent to \(|x|\), so this does not affect the graph itself.

Step 3: Reflect Across the X-Axis

The negative sign outside of the absolute value \(-|-x|\) simplifies to \(-|x|\), indicating a reflection across the x-axis. This flips the V-shape to open downward.

Step 4: Vertical Shrink

The coefficient \(-\frac{2}{5}\) represents a vertical shrink by a factor of \(\frac{2}{5}\). This makes the V-shape of the graph steeper.

Step 5: Vertical Shift

The term \(+6\) indicates a vertical shift upward by 6 units. This moves the entire graph up by 6 units.

Key Concepts

Absolute Value Function

The absolute value function, represented by the equation \( y = |x| \), is a cornerstone of graph transformations. Its graph is V-shaped, with the vertex located at the origin (0,0). This shape is symmetrical about the y-axis and opens upwards in its basic form. Understanding this function is key to applying various transformations.

When you see an absolute value function, it's important to recognize its core properties:

• Vertex: The point where the graph changes direction; always at (0,0) in the standard form.
• Symmetry: The graph is mirrored along the y-axis.
• Upward Opening: In the form \( y = |x| \), the "arms" of the V open upwards.

These characteristics serve as a baseline when applying transformations that modify the V-shape in different ways, such as reflections, vertical shifts, and vertical stretches or shrinks.

Reflection

Reflection is a key transformation in graphing that flips the graph over a specific axis. For the function \( y = -|x| \), the negative sign outside the absolute value symbol indicates a reflection across the x-axis. This transformation turns the normally upward-opening V shape of the absolute value function into an inverted V that opens downward.

Examining reflections involves two main axes:

• Reflection over the x-axis: Achieved by multiplying the function by -1 (i.e., \( - |x| \)), it flips the graph vertically.
• Reflection over the y-axis: While \( |-x| \) and \( |x| \) are equivalent for the absolute value function, it doesn't impact the shape.

Reflections fundamentally alter the orientation of a graph, providing a powerful tool for modifying the direction in which the arms of the V shape point.

Vertical Shrink

A vertical shrink or compression occurs when the graph of a function is scaled down relative to the y-axis. In the equation \( y = -\frac{2}{5}|x| \), the coefficient \(-\frac{2}{5}\) demonstrates this concept. Instead of stretching the graph vertically, it pulls it towards the x-axis, creating a steeper angle than the original V-shape.

Understanding vertical shrinks involves the following:

• A coefficient between 0 and 1 (or its negative counterpart) compresses the graph.
• The effect is observed in how quickly or steeply the graph rises or falls as it moves away from the origin.

This transformation can be contrasted with vertical stretches, where coefficients greater than 1 elongate the graph. Vertical shrinks adjust the steepness of the absolute value function's arms, changing how sharply the graph extends from its vertex.

Vertical Shift

Vertical shifts alter the position of the graph up or down along the y-axis. In the function \( y = -\frac{2}{5}|x| + 6 \), the \(+6\) indicates a vertical shift upwards by 6 units. This moves the entire graph upwards without altering its shape.

Key aspects of vertical shifts include:

• The addition or subtraction of a constant to the function shifts the entire graph.
• Addition results in an upward movement, whereas subtraction results in a downward shift.
• The vertex of the absolute value function, initially at (0,0), moves to (0,6) in our example.

Vertical shifts are straightforward yet impactful, affecting the baseline or starting point of the graph while maintaining the overall structure established by other transformations.