/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 The graph of \(y=f(x)\) passes t... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 87

The graph of \(y=f(x)\) passes through the points (0,1) \((1,2),\) and \((2,3) .\) Find the corresponding points on the graph of \(y=f(x+2)-1\).

Short Answer

Expert verified
The corresponding points on the graph of \(y=f(x+2)-1\) are \((-2,0), (-1,1), (0,2)\)

Step by step solution

01

Calculate the new x-coordinates

To calculate the new x-coordinates, subtract 2 from the original x-coordinates. \n This gives: \n For (0,1), new x-coordinate = \(0-2 = -2\) \n For (1,2), new x-coordinate = \(1-2 = -1\) \n For (2,3), new x-coordinate = \(2-2 = 0\)
02

Calculate the new y-coordinates

To calculate the new y-coordinates, subtract 1 from the original y-coordinates. \n This gives: \n For (0,1), new y-coordinate = \(1-1 = 0\) \n For (1,2), new y-coordinate = \(2-1 = 1\) \n For (2,3), new y-coordinate = \(3-1 = 2\)
03

Record the new points

Apply the new x and y coordinates to get the new points. The new points on the graph of \(y=f(x+2)-1\) corresponding to (0,1), (1,2), and (2,3) are \((-2,0), (-1,1), (0,2)\) respectively.

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Key Concepts

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

Graph Shifting
Graph shifting is a simple way to change the position of a function's graph on the coordinate plane. This includes moving the graph up or down, as well as left or right.
For instance, if you're given a function, say \(y = f(x)\), and you want to shift it, you make changes to the function formula.
  • Shifting Left or Right: To move a graph to the left, you add to the \(x\) value inside the function. To shift it to the right, you subtract from the \(x\) value inside the function. For example, \(y = f(x+2)\) shifts the graph two units to the left.
  • Shifting Up or Down: To shift the graph up, you add to the whole function. To shift it down, you subtract from the whole function. For instance, \(y = f(x) - 1\) moves the graph one unit down.
In our problem, the function is shifted left by 2 units due to the \(f(x+2)\) part and down by 1 unit because of the \(-1\) part.
Coordinates Transformation
Understanding how coordinates change under a transformation is key to mastering function transformations. This involves changing the \(x\) or \(y\) coordinates of points on the graph.
When you apply a transformation such as \(y = f(x+2) - 1\), each point on the graph undergoes a specific change:
  • The \(x\)-coordinates are affected by the "inside" the function changes \((+2)\), meaning you subtract 2 from each original \(x\)-coordinate to get the new ones.
  • The \(y\)-coordinates are affected by changes "outside" the function, which is the \(-1\) in this case, meaning you subtract 1 from each original \(y\)-coordinate.
Using this understanding on the points \((0,1), (1,2), (2,3)\) results in new points: \((-2,0), (-1,1), (0,2)\). This shows how each coordinate pair shifts together.
Function Graphing
Function graphing is the process of displaying a function on the coordinate plane, using points derived from coordinates.
To effectively graph the transformed function \(y = f(x+2) - 1\), it's essential to know the new sets of points. You take points from the original graph and apply the transformations as guided:
  • Begin with each point \((x, y)\) from the original function graph.
  • Adjust the \(x\)-coordinates for any shifts left or right, by subtracting or adding to them.
  • Adjust the \(y\)-coordinates for any upward or downward shifts by either adding or subtracting as required.
After transforming, plot the new points on the graph. In this exercise, it results in points \((-2,0), (-1,1), (0,2)\). Connecting these points as smoothly as possible will give you the graph of the new transformed function. Function graphing, in essence, allows visual representation of how algebraic expressions transform visually.

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Most popular questions from this chapter

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=10 $$

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

The total sales (in billions of dollars) for CocaCola Enterprises from 2000 through 2007 are listed below. (Source: Coca-Cola Enterprises, Inc.) $$\begin{array}{llll} 2000 & 14.750 & 2004 & 18.185 \\ 2001 & 15.700 & 2005 & 18.706 \\ 2002 & 16.899 & 2006 & 19.804 \\ 2003 & 17.330 & 2007 & 20.936 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the total revenue (in billions of dollars) and let \(t=0\) represent 2000 . (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the sales of Coca-Cola Enterprises in 2008 . (f) Use your school's library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e).

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=2 $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{8 x-4}{2 x+6} $$
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