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Chapter 1: Problem 48

Writing an Equation from a Description In Exercises \(47-54\) , write an equation for the function described by the given characteristics. The shape of \(f(x)=x^{2},\) but shifted two units to the left, nine units up, and then reflected in the \(x\) -axis

Short Answer

Expert verified
The equation of the function after the transformations is \(f(x) = -((x+2)^{2} + 9)\).

Step by step solution

01

Horizontal shift

A horizontal shift of a function can be represented by substituting \(x\) with \((x-h)\) in the function, where \(h\) represents the magnitude of the shift. When \(h>0\), the shift is to the right and when \(h<0\), the shift is to the left. Here, the function is shifted two units to the left. So, the function after the horizontal shift becomes \(f(x) = (x+2)^{2}\).
02

Vertical shift

A vertical shift of a function can be represented by adding a constant \(k\) to the function, where \(k\) represents the shift's magnitude. If \(k>0\), the shift is upward, and if \(k<0\), the shift is downward. Here, the function is shifted nine units up, so the function after the vertical shift becomes \(f(x) = (x+2)^{2} + 9\).
03

Reflection over x-axis

A reflection of a function about the x-axis can be represented by multiplying the function by -1. When the function is reflected about the x-axis, we obtain \(f(x) = -((x+2)^{2} + 9) \).

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Key Concepts

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

Horizontal Shift
A horizontal shift in function transformation involves moving the entire graph of a function a certain distance to the left or right without changing its shape.
To achieve this, you replace every instance of \(x\) in the function's formula with \((x-h)\). Here, \(h\) indicates the number of units the graph should move.
  • If \(h > 0\), the function shifts to the right by \(h\) units.
  • If \(h < 0\), the function shifts to the left by \(h\) units.
In our current example, we are looking to move the basic quadratic function \(f(x)=x^2\) two units to the left. Thus, we use \((x+2)\), indicating the shift is two units towards negative \(x\)-axis, leading to the function \(f(x)=(x+2)^2\). This concept is vital in understanding how changes along the x-axis affect the positioning of graphs.
Vertical Shift
A vertical shift involves moving the graph of a function up or down. Unlike horizontal shifts, which affect the x-coordinate, vertical shifts change the y-coordinate of each point on the graph.
To apply a vertical shift, a constant \(k\) is added to the entire function:
  • If \(k > 0\), the graph moves upwards by \(k\) units.
  • If \(k < 0\), the graph moves downwards by \(k\) units.
For instance, in our modification of \(f(x)=(x+2)^2\), the graph is shifted 9 units up.
This height change is depicted by adding 9 to the function, resulting in \(f(x)=(x+2)^2 + 9\). This operates independently of any horizontal shifting actions, solely affecting the vertical position.
Reflection in x-axis
Reflecting a graph over the x-axis inverts it vertically. This means that every point on the function's graph at \((x, y)\) moves to \((x, -y)\).
This transformation is represented mathematically by multiplying the function by \(-1\). Thus, if your current function is \(f(x)\), its reflection across the x-axis is \(-f(x)\).
  • This alteration flips the graph, maintaining its shape while reversing its orientation in the vertical plane.
In the example given, the function \(f(x) = (x+2)^2 + 9\) is reflected over the x-axis to produce \(-((x+2)^2 + 9)\).
This reflection seamlessly combines with the horizontal and vertical shifts applied earlier, completing the full transformation of the function.

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Most popular questions from this chapter

One-to-One Function Representation In Exercises 105 and \(106,\) determine whether the situation could be represented by a one-to-one function. If so, then write a statement that best describes the inverse function. The number of miles \(n\) a marathon runner has completed in terms of the time \(t\) in hours

Matching and Determining Constants In Exercises \(85-88\) , match the data with one of the following functions $$ f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \text { and } r(x)=\frac{c}{x} $$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-1} & {0} & {1} & {4} \\\ \hline y & {6} & {3} & {0} & {3} & {6} \\ \hline\end{array} $$

Think About It Consider the functions \(f(x)=x+2\) and \(f^{-1}(x)=x-2 .\) Evaluate \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x))\) for the indicated values of \(x .\) What can you conclude about the functions?

Graphical Reasoning Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. $$\begin{array}{ll}{f(x)=x^{2}-x^{4}} & {g(x)=2 x^{3}+1} \\ {h(x)=x^{5}-2 x^{3}+x} & {j(x)=2-x^{6}-x^{8}} \\ {k(x)=x^{5}-2 x^{4}+x-2} & {p(x)=x^{9}+3 x^{5}-x^{3}+x}\end{array}$$ What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. If the graph of the parent function \(f(x)=x^{2}\) is shifted six units to the right, three units up, and reflected in the \(x\) -axis, then the point \((-2,19)\) will lie on the graph of the transformation.
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