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Chapter 0: Problem 16

Sketch the graph of \(g(x)=|x+3|-4\) by first sketching \(h(x)=|x|\) and then translating.

Short Answer

Expert verified
Graph is a V-shape with vertex at (-3, -4).

Step by step solution

01

Understand the Parent Function

The function given is based on the absolute value function, which has the parent function: \( h(x) = |x| \). The graph of this function is a V-shaped curve, which intersects the y-axis at 0 and opens upwards with a vertex at the origin (0,0).
02

Identify the Transformations

The function \( g(x) = |x + 3| - 4 \) involves two transformations of the parent function \( h(x) = |x| \). First, \( |x+3| \) implies a horizontal shift of 3 units to the left. Second, subtracting 4 in \( -4 \) indicates a vertical shift of 4 units downwards.
03

Apply the Horizontal Translation

To incorporate the horizontal translation, move the vertex of \( h(x) = |x| \) from \( (0, 0) \) to \( (-3, 0) \). This represents the function \( k(x) = |x + 3| \), which is the first step in creating the graph of \( g(x) \).
04

Apply the Vertical Translation

Next, translate the graph of \( k(x) = |x+3| \) vertically downward by 4 units. Move the vertex from \( (-3, 0) \) to \( (-3, -4) \). The result is the graph of \( g(x) = |x+3| - 4 \).
05

Sketch the Final Graph

Draw the translated graph starting with the new vertex at \( (-3, -4) \). The graph maintains the same V-shape as the original absolute value function. It has two linear arms extending from the vertex with slopes of 1 and -1, reflecting how the graph diverges from the vertex points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Absolute Value Function
The absolute value function is a fundamental concept in mathematics. At its core, it measures the distance of a number from zero on the number line, ignoring negative signs. When graphed, the absolute value function, represented as \( h(x) = |x| \), forms a distinctive V-shaped curve.
  • This graph has its vertex at the origin, \((0, 0)\).
  • It features a slope of 1 upwards on the right side and a slope of -1 on the left side.
The absolute value function is crucial because it serves as a parent function for a wide range of transformations, just like an outline or a skeleton for more complex shapes. This simplicity allows it to serve as a starting point before applying transformations such as translations that further modify its shape and position.
Horizontal Translation: Shifting Left or Right
A horizontal translation involves shifting a graph left or right along the x-axis. This type of transformation does not alter the shape of the graph, only its position horizontally. For an absolute value function, a horizontal translation is achieved by adding or subtracting a constant within the absolute value expression.
In the function \(g(x) = |x+3|\), the expression \(x+3\) implies a horizontal translation of 3 units to the left. Why to the left? Because we translate in the opposite direction of the sign inside the function.
  • Start with the vertex at \((0, 0)\) from \(h(x) = |x|\).
  • Move this point to \((-3, 0)\) to represent \(k(x) = |x + 3|\).
Horizontal translations can make complex graphs easier to sketch by simply shifting the starting position without altering the basic structure.
Vertical Translation: Moving Up or Down
A vertical translation shifts a graph up or down along the y-axis, modifying its vertical position without affecting its shape. For the equation \(g(x) = |x + 3| - 4\), the "-4" term shifts the entire graph 4 units down. This is known as a vertical displacement.
  • Start from \((-3, 0)\), the vertex obtained from the horizontal translation.
  • Shift this vertex to \((-3, -4)\) to get the final graph of \(g(x)\).
Much like adjusting a picture hanging on the wall, a vertical translation changes where the graph 'sits' vertically without touching its form. This makes the process of graphing translations intuitive, as you can independently adjust horizontal and vertical positions.

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