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Chapter 2: Problem 34

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)=2|x+3| $$

Short Answer

Expert verified
The graph of the function \(h(x)=2*|x+3|\) is a V-shaped graph which is obtained by applying a vertical stretch of factor 2 to the graph of the absolute value function \(f(x) = |x|\) and then shifting it to the left by 3 units.

Step by step solution

01

Graph the base function \(f(x) = |x|\)

In order to graph \(f(x) = |x|\), start by plotting points. The graph of \(f(x) = |x|\) is V-shaped with the point (0,0) at the apex of the V. For \(x>0\), the graph coincides with the line \(y = x\), and for \(x<0\) it coincides with the line \(y = -x\).
02

Apply the vertical stretch

The vertical stretch is determined by the multiplier '2'. Apply the vertical stretch to the graph by simply multiplying the y-values of the graph by 2.
03

Apply the horizontal shift

The |x+3| in the function \(-h(x)=2*|x+3|\) implies a shift to the left by 3 units. Simply move every point on your graph three units to the left.

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Key Concepts

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

Absolute Value Functions
Absolute value functions are an interesting type of mathematical function that can be identified by their distinctive V-shaped graphs. The most basic form of an absolute value function is written as \(f(x) = |x|\).
The absolute value symbol |x| represents the distance of x from zero on a number line, ignoring any negative sign.
This means when you graph \(f(x) = |x|\), you see a V shape that starts at the origin point (0,0).
  • For x values greater than zero, the graph follows the line \(y = x\).
  • For x values less than zero, the graph follows the line \(y = -x\).

This reflects the idea that the absolute value is always positive or zero.
The point at the origin is called the "vertex" of the absolute value graph.
Vertical Stretch
A vertical stretch affects how steep or flat the graph of a function appears. When we talk about a vertical stretch, we refer to multiplying the y-values of a function by a certain factor.
In our case, the function \(h(x) = 2|x+3|\) involves a vertical stretch by a factor of 2.
This means that every point on the basic absolute value graph \(f(x) = |x|\) will have its y-value doubled.
  • If a point on the original graph is at \((x, y)\), the new position will be \((x, 2y)\).
  • This transformation causes the whole V shape to become narrower, making it appear 'stretched' vertically.
  • The effect is similar to pulling the arms of the V upwards without changing their direction.

It's important to note that a vertical stretch affects only the y-values and doesn鈥檛 shift the graph left or right.
Horizontal Shift
A horizontal shift moves the graph of a function left or right along the x-axis.
With our function \(h(x) = 2|x+3|\), we encounter a horizontal shift because of \(x+3\).
This expression tells us to move the entire graph of the absolute value function left by 3 units.
  • The key is to understand that adding a number inside the absolute value causes a shift in the opposite direction of the sign.
  • By moving the graph left 3 units, each point on the original graph \(f(x) = |x|\) is adjusted to \((x-3, y)\).
  • This transformation alters the vertex, shifting it from (0, 0) to (-3, 0).

Horizontal shifts never change the shape of the graph; they only affect the position horizontally.
Graphing Techniques
Mastering graphing techniques is essential for transforming and understanding functions visually.
For absolute value functions, these techniques can greatly simplify graphing tasks. Here鈥檚 a simple process to follow:
  • Begin by plotting the original basic function. In our case, \(f(x) = |x|\) with your familiar V shape.
  • Apply any vertical transformations next, like stretches or compressions, which in our example multiplies all y-values by 2 moving onto \(g(x) = 2|x|\).
  • Then proceed with horizontal transformations, such as shifts. For \(h(x) = 2|x+3|\), this means shifting the graph left by 3 units.
  • Check your work by identifying new coordinates: the vertex moves from (0, 0) to (-3, 0), and each arm of the V adjusts accordingly.

Practice makes perfect鈥攕o try graphing several functions with different transformations to gain confidence. This methodical approach helps you tackle any graphing transformation challenge.

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Most popular questions from this chapter

Almanacs, newspapers, magazines, and the Internet contain bar graphs and line graphs that describe how things are changing over time. For example, the graphs in Exercises \(79-82\) show how various phenomena are changing over time. Find a bar or line graph showing yearly changes that you find intriguing. Describe to the group what interests you about this data. The group should select their two favorite graphs. For each graph selected: a. Rewrite the data so that they are presented as a relation in the form of a set of ordered pairs. b. Determine whether the relation in part (a) is a function. Explain why the relation is a function, or why it is not.

What is the slope of a line and how is it found?

Which one of the following is true? a. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of three units to the right and three units upward of the graph of \(f\) b. If \(f(x)=-\sqrt{x}\) and \(g(x)=\sqrt{-x},\) then \(f\) and \(g\) have identical graphs. c. If \(f(x)=x^{2}\) and \(g(x)=5\left(x^{2}-2\right),\) then the graph of \(g\) can be obtained from the graph of \(f\) by stretching \(f\) five units followed by a downward shift of two units. d. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting in the \(x\) -axis, and then moving the resulting graph down four units.

You will be developing functions that model given conditions. A company that manufactures bicycles has a tixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, \(C\), as a function of the number of bicycles produced. Then find and interpret

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.
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