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Chapter 2: Problem 59

What must be done to a function's equation so that its graph is shifted vertically upward?

Short Answer

Expert verified
To shift the graph of a function vertically upward, a positive constant should be added to the function's equation.

Step by step solution

01

Understanding Shifting of Function Graphs

Before we proceed to the operation, let's understand how shifting occurs in the graphs of functions. A vertical shift up or down happens when a constant is added or subtracted from a function.
02

Upwards Shift Operation

For an upward shift, we add a positive constant to the function. Hence, if we have a function denoted by \(f(x)\), to shift its graph upwards by a units, we transform it into \(f(x) + a\).

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Most popular questions from this chapter

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. 90\. \(T(70,000)\)

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x)\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=(x-1)^{2} $$

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=|x|+4 $$

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=2(x-2)^{2} $$
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