Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x+4|-2$$

Step-by-Step Solution

Answer

The graph is a V-shape, shifted left 4 units and down 2 units.

Step 1: Identify the Base Function

The given function is \( y = |x+4| - 2 \). The basic function here is \( y = |x| \), which is an absolute value function. Our task is to apply transformations to \( y = |x| \) to obtain \( y = |x+4| - 2 \).

Step 2: Apply Horizontal Translation

The expression inside the absolute value, \( x+4 \), represents a horizontal translation. \( y = |x+4| \) shifts the graph of \( y = |x| \) 4 units to the left. Adjust the base function's vertex from \( (0, 0) \) to \( (-4, 0) \).

Step 3: Apply Vertical Translation

Next, consider the \(-2\) outside the absolute value, \( y = |x+4| - 2 \). This indicates a vertical translation, moving the graph of \( y = |x+4| \) down by 2 units. Shift the new vertex from \( (-4, 0) \) to \( (-4, -2) \).

Step 4: Sketch the Graph

Start by drawing the basic shape of \( y = |x| \), which is a V-shaped graph. Shift this graph 4 units to the left and 2 units down to reflect the translations. The vertex of the graph will be at \( (-4, -2) \), and the arms of the V will open upwards, following the pattern of \( y = |x| \).

Step 5: Verify the Translations

To ensure accuracy, pick a few points around the vertex, such as \( x = -5 \) and \( x = -3 \), and calculate their corresponding \( y \) values using \( y = |x+4| - 2 \). For \( x = -5 \), \( y = |-5+4| - 2 = |−1| - 2 = 1 - 2 = -1 \). For \( x = -3 \), \( y = |−3+4| - 2 = |1| - 2 = 1 - 2 = -1 \). Check these points to confirm the graph's shape.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( y = |x| \), is a fundamental mathematical function characterized by its signature V-shape. The absolute value of a number is its distance from zero on the number line, producing a non-negative result. This means the graph is symmetric around the y-axis and always opens upwards, resembling a V.

• The vertex of this graph is located at the origin \((0,0)\).
• The arms of the V extend in both directions, forming the two straight lines of the graph.
• For \( y = |x| \), when \( x \) is positive or zero, \( y = x \), and when \( x \) is negative, \( y = -x \).

Understanding the properties of the absolute value function is essential for graph sketching, as it serves as the base graph for transformations.

Function Transformation

Function transformations involve altering the graph of a base function in various ways to achieve a new function graph. These transformations can include shifting, stretching, compressing, or reflecting the graph.

• Translation: Moving the entire graph horizontally or vertically without changing its shape.
• Scaling: Stretching or compressing the graph vertically or horizontally, affecting its steepness or width.
• Reflection: Flipping the graph across an axis.

In the case of the function \( y = |x+4| - 2 \), transformations involve horizontal and vertical translations to modify the base graph \( y = |x| \). Understanding these transformations helps in manually sketching graphs without the need for technology.

Horizontal Translation

A horizontal translation shifts the entire graph of a function left or right along the x-axis. This type of transformation changes only the x-coordinates of the graph's points while maintaining the same shape and orientation.

• The transformation \( y = |x+4| \) means the graph of \( y = |x| \) is moved left by 4 units.
• To determine the direction of movement, examine the expression inside the absolute value. If it's \( x+h \), move left by \( h \) units; if it's \( x-h \), move right by \( h \) units.
• Adjust the vertex from \((0, 0)\) to \((-4, 0)\) as a result of this translation.

Being clear about the direction and magnitude of horizontal translations helps accurately sketch and predict the position of the graph after transformation.

Vertical Translation

A vertical translation involves moving the graph up or down along the y-axis. This transformation affects the y-coordinates of each point on the graph while keeping the x-values constant.

• For \( y = |x+4| - 2 \), the graph undergoes a vertical translation downward by 2 units.
• This is indicated by the \(-2\) outside the absolute value function, which directly shifts the graph down.
• The vertex, after the horizontal translation to \((-4, 0)\), moves to \((-4, -2)\) following the vertical translation.

Understanding vertical translations is necessary for accurately positioning the graph after all transformations, ensuring points are plotted correctly relative to their base function position.