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Chapter 2: Problem 129

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

Short Answer

Expert verified
To vertically shift a function's graph upward by 'a' units, simply add 'a' to the function. Thus, if the starting equation was \(f(x)\), the transformed function would be \(f(x) + a\).

Step by step solution

01

Understanding Function Transformations

In algebra, function transformations involve making changes to the original function, such as shifting, stretching, or flipping the function. In this particular case, we're focusing on a vertical shift, which means we are moving the entire graph up or down without changing its shape.
02

Shifting a function vertically

A function \(f(x)\) is shifted vertically upwards by a units when a constant 'a' is added to it, resulting in a new function denoted as \(g(x) = f(x) + a\). This change suggests that for every x-value, the y-value of the function will be a units higher than the original function.
03

Example of Vertical Shift

Consider a function \(f(x) = x^2\). If we want to shift this function upwards by 3 units, we add 3 to the function to get our new function: \(g(x) = x^2 + 3\). This has the effect of moving every point on the original graph upwards by 3 units.

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Most popular questions from this chapter

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=|3 x-4|$$

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=\frac{1}{4 x+5}$$

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=|2 x-5|$$

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\frac{x}{x+5}, g(x)=\frac{6}{x}$$

Exercises \(103-105\) will help you prepare for the material covered in the next section. Let \(\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.
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