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Chapter 2: Problem 52

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function g using trans/ations and/or a reflection. See Examples 7 and 8 . $$ g(x)=|x+4|+3 $$

Short Answer

Expert verified
Sketch a 'V' shape moved left to (-4,3).

Step by step solution

01

Recognize Base Function

Identify the base function from the given functions. The base function for \( g(x) = |x+4| + 3 \) is \( f(x) = |x| \). This function represents a 'V' shape centered at the origin (0,0) and always outputting non-negative values.
02

Sketch the Base Function

Draw the graph of \( f(x) = |x| \). It is a 'V' shape with the vertex at the origin (0,0), opening upwards, where \( f(x) = |x| \) equals to \( x \) for \( x \geq 0 \) and \( -x \) for \( x < 0 \).
03

Determine Horizontal Translation

Analyze the expression \( |x+4| \) in \( g(x) = |x+4| + 3 \). The \( +4 \) inside the absolute value indicates a horizontal shift to the left by 4 units. This moves the vertex of the 'V' shape from (0,0) to (-4,0).
04

Determine Vertical Translation

The expression \( +3 \) outside the absolute value in \( g(x) = |x+4| + 3 \) indicates a vertical shift upwards by 3 units. This moves the vertex from (-4,0) to (-4,3).
05

Sketch the Transformed Function

Draw the graph of \( g(x) = |x+4| + 3 \) by shifting the vertex of the 'V' shape of \( f(x) = |x| \) from (0,0) to (-4,3). The graph keeps its 'V' shape, opening upwards, intersecting the y-axis at (0,7).

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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 \( f(x) = |x| \), is an important and fundamental function in algebra and calculus. At its core, the absolute value measures how far a number is from zero on the number line, regardless of direction.

Its graph is distinctive, forming a "V" shape, opening upwards. This shape appears because the function
  • equals \( x \) when \( x \) is greater than or equal to zero
  • is \( -x \) when \( x \) is less than zero

The point where the "arms" of the "V" meet is called the vertex, and it's located at the origin (0,0) for the basic absolute value function. The absolute value function is a simple yet powerful tool because it transforms negative values into positive ones.
Vertical Translation
Vertical translation involves moving a graph up or down without changing its shape. In the case of the equation \( g(x) = |x+4| + 3 \), the \(+3\) outside the absolute value function indicates that we need to lift the entire graph upwards by 3 units.

This operation affects every point on the graph, moving all points higher by the same amount. Thus, although the shape of the function remains a "V", the vertex initially at \((-4,0)\) shifts to \((-4,3)\).

Such transformations are quite intuitive:
  • If the number added is positive, the graph moves up.
  • If it is negative, the graph will move down.
This principle allows us to easily adjust the vertical position of any function to fit our needs.
Horizontal Translation
Whenever we see an expression like \(|x+4|\) in a function, we're dealing with a horizontal translation. This changes the graph's position horizontally on the coordinate plane.

For the function \( g(x) = |x+4| + 3 \), the \(+4\) inside the absolute value suggests moving the entire graph 4 units to the left. This is opposite to what one might initially think, as it shifts left instead of right.

Here鈥檚 why:
  • When there's \(+c\) inside \(|x+c|\), shift left by \(c\) units.
  • When there's \(-c\), shift right by \(c\) units.
Such translations are useful because they allow us to center graphs wherever we need them, without altering their shape.
Reflection in Graphs
Reflections in graphs involve flipping a graph over a specified axis, such as the x-axis or y-axis. While reflections aren't directly involved in the function \( g(x) = |x+4| + 3 \), it's a concept worth understanding for transforming graphs.

Reflections can be visualized as placing a mirror on the axis you are reflecting across:
  • A reflection across the x-axis takes a graph and inverts it from up to down or vice versa.
  • Reflecting across the y-axis mirrors the graph from right to left.
Reflections are valuable for creating symmetries and altering the orientation of graphical shapes in a variety of functions.

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Find \(g(2)\) and \(g(3)\). \(g(x)=x^{3}-x\)
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