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Chapter 1: Problem 33

Starting with the graph of \(f(x)=6^{x}\) write the equation of the graph that results from a. reflecting \(f(x)\) about the \(x\) -axis and the \(y\) -axis b. reflecting \(f(x)\) about the \(x\) -axis, shifting left 2 units, and down 3 units

Short Answer

Expert verified
a. \( f(x) = -6^{-x} \); b. \( f(x) = -6^{x+2} - 3 \)

Step by step solution

01

Reflecting Over the x-axis

To reflect the graph of the function over the x-axis, we multiply the function by -1. Thus, the equation of the graph after this reflection becomes \( f(x) = -6^x \).
02

Reflecting Over the y-axis

To reflect the graph over the y-axis, we replace \( x \) with \( -x \) in the function. Therefore, the new equation is \( f(x) = 6^{-x} \).
03

Reflecting Over Both Axes

To reflect over both axes, we apply both transformations from the previous steps together. Replacing \( x \) with \( -x \) and multiplying by -1 gives the equation \( f(x) = -6^{-x} \).
04

Shifting Left 2 Units

Shifting a graph to the left involves adding a constant to the \( x \) value. For a left shift of 2 units, replace \( x \) with \( x+2 \) in the function. The equation becomes \( f(x) = -6^{x+2} \).
05

Shifting Down 3 Units

To shift a graph down 3 units, subtract 3 from the function. The final equation is \( f(x) = -6^{x+2} - 3 \).

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Key Concepts

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

Reflection of Functions
Reflecting a function can be understood as flipping it over a specific axis. This creates a mirror image of the original graph.
  • Reflection over the x-axis: To reflect a function over the x-axis, you multiply the entire function by -1. If your original function is \( f(x) = 6^x \), then its reflection will be \( f(x) = -6^x \). This transformation flips the graph upside down, affecting all the y-values of the function.
  • Reflection over the y-axis: Here, the procedure involves replacing \( x \) with \( -x \) in the function. Taking \( f(x) = 6^x \) as an example, the reflection over the y-axis will be \( f(x) = 6^{-x} \). This transformation changes the direction of the graph along the x-axis but keeps the y-values unchanged.
  • Reflection over both axes: By combining the two reflections described above, you reflect the function over both axes. This means applying both transformations: replacing \( x \) with \( -x \) and multiplying the entire function by -1. The result for our example is \( f(x) = -6^{-x} \).
Understanding reflections is crucial, as they help in visualizing how graphs can transform into their mirrored counterparts.
Shifting Functions
Shifting, or translating a function involves moving the entire graph a certain number of units in a specified direction. This does not alter the shape of the graph, only its position.
  • Horizontal Shifts: Moving the graph to the left or right involves changing the \( x \) values. For example, to shift \( f(x) = 6^x \) 2 units to the left, you replace \( x \) with \( x + 2 \) in the function. Thus, we get \( f(x) = 6^{x+2} \).
  • Vertical Shifts: This shift involves moving the graph up or down by changing the y-values. For shifting down, you simply subtract a constant from the function. For instance, shifting \( f(x) = 6^x \) down by 3 units results in \( f(x) = 6^x - 3 \).
Shifting is a fundamental transformation that alters the location of the graph on the coordinate plane but does not affect its general shape. It's as if you're sliding the graph to fit a new space on the Cartesian plane.
Exponential Functions
Exponential functions are functions where the variable is in the exponent. They are defined as \( f(x) = a^x \), where \( a \) is a constant and \( a > 0 \).
  • Characteristics: Exponential functions have a rapid growth rate with positive bases or decay rate with fractional bases. The base, \( a \), determines the rapidity of the growth or decay. For example, \( f(x) = 6^x \) grows quickly as \( x \) increases.
  • Graphing Exponential Functions: The graph of an exponential function is a continuous curve. It's crucial to note that when \( a > 1 \), the graph rises, whereas if \( 0 < a < 1 \), it falls as \( x \) increases. With \( f(x) = 6^x \), the function grows swiftly, making it an example of exponential growth.
Exponential functions are widely used in modeling situations where quantities grow or decay rapidly, such as populations, radioactive decay, and finance scenarios. Understanding their behavior helps in grasping the impact of exponential growth or decay in various real-world contexts.

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Most popular questions from this chapter

Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down. $$ g(t)=t \sqrt{t+3} $$

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ p(x)=\left(\frac{1}{3} x\right)^{2}-3 $$

Write a formula for the function that results when the given toolkit function is transformed as described. \(f(x)=\frac{1}{x}\) vertically stretched by a factor of \(8,\) then shifted to the right 4 units and up 2 units.

Select all of the following tables which represent \(y\) as a function of \(x\) and are one-to-one. a. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 8 & 8 \\ \hline \boldsymbol{y} & 5 & 6 & 13 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|l|l|} \hline x & 2 & 8 & 14 \\ \hline y & 5 & 6 & 6 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 8 & 14 \\ \hline \boldsymbol{y} & 5 & 6 & 13 \\ \hline \end{array} $$

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ n(x)=\frac{1}{3}|x-2| $$
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