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Chapter 0: Problem 7

Graph each pair of equations on one set of axes. $$y=|x| \text { and } y=|x-3|$$

Short Answer

Expert verified
The graph of \(y = |x|\) is a 'V' shape with vertex at (0,0), and \(y = |x - 3|\) is a 'V' shape with vertex at (3,0).

Step by step solution

01

Understand the Equations

The first equation is given as \(y = |x|\) which represents an absolute value function. This function creates a 'V' shape with its vertex at the origin (0,0). The second equation is given as \(y = |x - 3|\), which is also an absolute value function but shifted to the right by 3 units.
02

Identify Key Points for y = |x|

Identify key points for \(y = |x|\). When \(x = 0\), \(y = 0\). For \(x > 0\), \(y = x\) and for \(x < 0\), \(y = -x\). Key points include (0,0), (1,1), (2,2), (-1,1), (-2,2).
03

Graph y = |x|

Plot the points (0,0), (1,1), (2,2), (-1,1), (-2,2) on the graph. Connect them to form a 'V' shape with the vertex at (0,0).
04

Identify Key Points for y = |x - 3|

Identify key points for \(y = |x - 3|\). When \(x = 3\), \(y = 0\). For \(x > 3\), \(y = x - 3\) and for \(x < 3\), \(y = 3 - x\). Key points include (3,0), (4,1), (5,2), (2,1), (1,2).
05

Graph y = |x - 3|

Plot the points (3,0), (4,1), (5,2), (2,1), (1,2) on the graph. Connect them to form a 'V' shape with the vertex at (3,0).
06

Analyze the Graphs

Observe that \(y = |x|\) has its vertex at (0,0) and \(y = |x - 3|\) has its vertex at (3,0). Both graphs form a 'V' shape and are symmetric. The graph of \(y = |x - 3|\) is the same as \(y = |x|\) but shifted 3 units to the right.

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Key Concepts

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

Graphing Absolute Value Functions
Absolute value functions create V-shaped graphs. When graphing the function, focus on key points and how they form this characteristic shape. For the equation, \(y = |x|\), it splits into two linear parts;
  • For \(x \geq 0\), \(y = x\)
  • For \(x < 0\), \(y = -x\)
These lines meet at the vertex, which is the lowest or highest point of the V-shape, located at the origin (0,0). You can plot points like (0,0), (1,1), (2,2), (-1,1), and (-2,2). Once plotted, they form the V shape with symmetry around the y-axis. Understanding how to plot these points accurately is essential for graphing absolute value functions.
Transformations of Functions
Transformations shift, stretch, or compress the graph. The equation \(y = |x - 3|\) shows a horizontal transformation of the base function \(y = |x|\). Specifically, it shifts the graph 3 units to the right. This happens because replacing \(x\) with \(x - 3\) means all x-values are effectively increased by 3. The key points now become:
  • (3,0) instead of (0,0)
  • (4,1) instead of (1,1)
  • (5,2) instead of (2,2)
  • (2,1) instead of (-1,1)
  • (1,2) instead of (-2,2)
The vertex, originally at (0,0), is now at (3,0). Transformations are crucial to understanding how basic functions change in different scenarios.
Vertex of Absolute Value Functions
The vertex is a key feature of absolute value functions. It represents the point where the two linear parts of the function meet. For the function \(y = |x|\), the vertex is at the origin (0,0). When transformations are applied, their new vertices can be found by looking at how the function is modified. In the equation \(y = |x - 3|\), the graph shifts horizontally. Consequently, the vertex moves from (0,0) to (3,0). Identifying the vertex helps in accurately drafting the V shape of the function, confirming the position and nature of the graph's minimal or maximal point.

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Most popular questions from this chapter

Boxowitz, Inc., a computer firm, is planning to sell a new graphing calculator. For the first year, the fixed costs for setting up the new production line are 100,000 dollars. The variable costs for producing each calculator are estimated at 20 dollars. The sales department projects that 150,000 calculators can be sold during the first year at a price of 45 dollars each. a) Find and graph \(C(x),\) the total cost of producing \(x\) calculators. b) Using the same axes as in part (a), find and graph \(R(x),\) the total revenue from the sale of \(x\) calculators. c) Using the same axes as in part (a), find and graph \(P(x),\) the total profit from the production and sale of \(x\) calculators. d) What profit or loss will the firm realize if the expected sale of 150,000 calculators occurs? e) How many calculators must the firm sell in order to break even?

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$y-7=x^{3}$$

Sally makes deposits into a retirement account every year from the age of 30 until she retires at age 65 . a) If Sally deposits 1200 per year and the account earns interest at a rate of \(8 \%\) per year, compounded annually, how much does she have in the account when she retires? (Hint: Use the annuity formula for Exercises 35 and 36 . ) b) How much of that total amount is from Sally's deposits? How much is interest?

Suppose that \(\$ 3000\) is borrowed as a college loan, at \(5 \%\) interest, compounded daily, for \(t\) years. a) The amount \(A\) that is owed is a function of time. Find an equation for this function. b) Determine the domain of the function in part (a).

Quick Copy buys an office machine for 5200 dollars on January 1 of a given year. The machine is expected to last for 8 yr, at the end of which time its salvage value will be 1100 dollars. If the company figures the decline in value to be the same each year, then the book value, \(V(t),\) after \(t\) years, \(0 \leq t \leq 8,\) is given by $$V(t)=C-t\left(\frac{C-S}{N}\right)$$ where \(C\) is the original cost of the item, \(N\) is the number of years of expected life, and \(S\) is the salvage value. a) Find the linear function for the straight-line depreciation of the office machine. b) Find the book value after 0 yr, 1 yr, 2 yr, 3 yr, 4 yr, 7 yr, and 8 yr.
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