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Magazine Chapter 2: Problem 53
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. $$y=|x+4|-2$$
Short Answer
Expert verified
The graph is a V-shape, shifted left 4 units and down 2 units.
Step by step solution
01
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 \( y = |x+4| - 2 \).
02
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 \( (-4, 0) \).
03
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 \( (-4, -2) \).
04
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 \( y = |x| \).
05
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 \( x = -5 \), \( y = |-5+4| - 2 = |鈭�1| - 2 = 1 - 2 = -1 \). For \( x = -3 \), \( y = |鈭�3+4| - 2 = |1| - 2 = 1 - 2 = -1 \). Check these points to confirm the graph's shape.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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, \( y = -x \).
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.
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.
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.
