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Chapter 3: Problem 83

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x)+2\)

Short Answer

Expert verified
Points: (-12, 8), (0, 10), (8, -2)

Step by step solution

01

Identifying Given Points

Identify the coordinates of points given on the graph of \(y = f(x)\). These points are \((-12, 6)\), \((0, 8)\), and \((8, -4)\).
02

Understanding the Transformation

Understand that \(g(x) = f(x) + 2\) means that to find \(y\) values on the graph of \(g(x)\), you add 2 to each corresponding \(y\) value from the graph of \(f(x)\).
03

Applying the Transformation to Each Point

1. Starting with \((-12, 6)\), the new \(y\) value will be \(6 + 2 = 8\). Thus, the corresponding point on \(y = g(x)\) is \((-12, 8)\).2. For \((0, 8)\), the new \(y\) value will be \(8 + 2 = 10\), giving the point \((0, 10)\).3. For \((8, -4)\), the new \(y\) value will be \(-4 + 2 = -2\), resulting in the point \((8, -2)\).

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Key Concepts

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

Translation of Graph
Graph translations involve shifting the entire graph of a function in one direction without changing its shape. A key type of transformation is the vertical shift, which is directly related to this exercise.
This alteration is expressed in the functional form as adding or subtracting a constant to the function. For example, in our problem, the function transformation is given by:
  • If you have a function \(f(x)\), and you define a new function \(g(x) = f(x) + 2\), this results in a vertical translation.
  • Every point on the graph of \(f(x)\) is shifted up by 2 units to form the graph of \(g(x)\).
The beauty of this transformation lies in its simplicity: it only affects the \(y\)-values of the points, leaving the \(x\)-values unchanged.
Coordinate Points
Coordinate points form the basis of understanding graphs and are essential in plotting. A coordinate point \((x,y)\) represents a location on the Cartesian plane. The \(x\) value, or abscissa, refers to the horizontal position, while the \(y\) value, or ordinate, indicates the vertical position.
  • In our exercise, we start with the given points:
    • \((-12, 6)\),
    • \((0, 8)\),
    • \((8, -4)\)
When we perform the vertical translation, only the \(y\)-values change. To compute the new points for the translated graph \(g(x) = f(x) + 2\):
  • \((-12, 6)\) becomes \((-12, 8)\)
  • \((0, 8)\) becomes \((0, 10)\)
  • \((8, -4)\) becomes \((8, -2)\)
These simple calculations are a direct reflection of understanding shifts in coordinate points, demonstrating how each point on a graph shifts based on the function's transformation method.
Graph of a Function
A graph of a function visually represents all the set points \((x, y)\) satisfying the function equation. It provides a mechanism to see the detailed behavior of a function across different values. For a clear example:
  • In the case of \(y = f(x)\), we initially plot the points such as \((-12, 6)\), representing the behavior of the original function.
As transformations like \(y = g(x)\), such as \(g(x) = f(x) + 2\), are applied, the graph changes accordingly. These adjustments enable new graphs to visualize shifts in function behavior.
  • After translation, new points, like \((-12, 8)\), are plotted, detailing the function's modified state.
This graphical approach helps students and teachers alike to intuitively understand and analyze how functions transform, offering a tangible way to perceive function relationships through shifted graphs.

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Most popular questions from this chapter

Use transformations of graphs to model the table of data with the formula \(f(x)=a(x-h)^{2}+k .\) (Answers may vary.) Average sales price of a home in thousands of dollars $$ \begin{array}{|cccccc} \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2005 \\ \hline \text { Price } & 30 & 80 & 150 & 210 & 300 \end{array} $$

Use transformations to sketch a graph of \(f\). \(f(x)=2+\sqrt{-(x-3)}\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 3 x^{2}+3=5 x $$

A window comprises a square with sides of length \(x\) and a semicircle with diameter \(x\) as shown in the figure. If the total area of the window is 463 square inches, estimate the value of \(x\) to the nearest hundredth of an inch. (PICTURE NOT COPY)

Solve. Write answers in standard form. $$ 3 x(3-x)-8=x(x-2) $$
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