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Chapter 1: Problem 16

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

Short Answer

Expert verified
Shift \(h(x)=|x|\) left by 3 and down by 4.

Step by step solution

01

Sketch the Graph of h(x)

To begin, let's consider the function \(h(x) = |x|\). This is a V-shaped graph that is symmetrical about the y-axis. At \(x = 0\), \(h(x) = 0\). The left side of the V decreases linearly with a slope of -1, and the right side of the V increases linearly with a slope of 1.
02

Translate Horizontally

Next, we need to translate the graph of \(h(x) = |x|\) horizontally. The function inside the absolute value part of \(g(x)\) is \(|x + 3|\). This indicates a translation to the left by 3 units. So, move each point on the graph of \(h(x)\) to the left by 3 units.
03

Translate Vertically

Finally, we need to incorporate the -4 outside the absolute value, \(|x + 3| - 4\). This will move each point on the graph 4 units down. Apply this vertical translation to the graph you obtained after the horizontal translation.
04

Draw the Final Graph

Now combine these transformations to sketch \(g(x) = |x + 3| - 4\). The vertex of the V-shaped graph has moved from (0, 0) to (-3, -4). Draw the V-shape with the same slope as \(h(x)\), but starting from the new vertex.

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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 that you often encounter in mathematics. In its basic form, the absolute value function is written as \( h(x) = |x| \). This function creates a V-shaped graph where the center or vertex is located at the origin of the coordinate plane, at the point (0, 0).
  • The left side of the graph has a slope of -1.
  • The right side has a slope of +1.
The absolute value function reflects the idea that the graph will mirror on both sides of the y-axis. This means whatever x-value you choose, whether negative or positive, will result in the same positive y-value, giving it its characteristic symmetrical V-shape.
Exploring Horizontal Translation
Horizontal translation shifts the graph of a function left or right. In our function \( g(x) = |x + 3| - 4 \), we see the term \( x+3 \) inside the absolute value. This signifies a horizontal translation of our graph.Horizontal shifts can be a bit tricky:
  • If it鈥檚 \( x + c \), you move the graph to the left by \( c \) units.
  • Conversely, if it鈥檚 \( x - c \), shift the graph to the right by \( c \) units.
In our example, we translate the graph of \( |x| \) to the left by 3 units. Imagine sliding the entire V-shape to the left on the x-axis, resulting in the new vertex position (-3, 0). This repositioning is crucial as it lays the foundation for further transformations.
Delving into Vertical Translation
Vertical translation involves shifting a graph up or down along the y-axis. Consider the function \( g(x) = |x + 3| - 4 \). The -4 that sits outside the absolute value indicates a downward vertical shift.This is easier to interpret:
  • When subtracting, it shifts the graph downward by the given value.
  • When adding, it shifts the graph upward by the given value.
In this scenario, after moving the graph left by 3 units, we shift it down by 4 units. This adjustment to the graph positions the vertex precisely at (-3, -4). The downward shift affects every point on the graph equally, maintaining the original V-shape but altering its location on the plane.
Putting It All Together with Function Graphing
Function graphing involves translating mathematical instructions into a visual graph. It鈥檚 like telling a story with numbers that become very clear when drawn. For \( g(x) = |x + 3| - 4 \), we have taken two main transformations:
  • A horizontal shift moving the graph left by 3 units.
  • A vertical shift moving the graph down by 4 units.
Combining these, the graph has the same shape and slopes as the original function \( h(x) = |x| \), but now its vertex is at (-3, -4). Graphing is a powerful tool, simplifying complex functions into understandable visual representations. This not only makes the problem easier to solve but also highlights the beauty of how functions can transform and move across the co-ordinate plane.

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Most popular questions from this chapter

Write an equation for the line through \((3,-3)\) that is (a) parallel to the line \(y=2 x+5\); (b) perpendicular to the line \(y=2 x+5\); (c) parallel to the line \(2 x+3 y=6\); (d) perpendicular to the line \(2 x+3 y=6\); (e) parallel to the line through \((-1,2)\) and \((3,-1)\); (f) parallel to the line \(x=8\); (g) perpendicular to the line \(x=8\).

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Convert the following degree measures to radians (leave \(\pi\) in your answer). (a) \(30^{\circ}\) (b) \(45^{\circ}\) \(370^{\circ}(\mathrm{c})\) (c) \(-60^{\circ}\) (d) \(240^{\circ}\) (e) \(-370^{\circ}\) (f) \(10^{\circ}\)
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