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Chapter 4: Problem 17

Sketch the graph of \(f\) $$f(x)=-\left(\frac{1}{2}\right)^{x}+4$$

Short Answer

Expert verified
Reflect the base graph across the x-axis and shift up by 4 units.

Step by step solution

01

Identify the base graph

Start by identifying the base graph, which is a simple exponential function. The base graph is given by \(g(x) = \left(\frac{1}{2}\right)^x\). This is a decreasing exponential function.
02

Apply the reflection

The function \(f(x) = -\left(\frac{1}{2}\right)^x+4\) includes a negative sign in front of the base function. This means reflect the graph of \(g(x)\) over the x-axis. If the original graph of \(g(x)\) was above the x-axis, the reflection will put it below the x-axis, maintaining its decreasing nature.
03

Vertical shift

The \(+4\) in \(f(x) = -\left(\frac{1}{2}\right)^x + 4\) indicates a vertical shift. Shift the entire reflected graph up by 4 units. This changes the horizontal asymptote from \(y = 0\) (the x-axis) to \(y = 4\).
04

Sketch the graph

Taking into account the reflection and vertical shift, draw the graph starting as a horizontal line near \(y = 4\) and decreasing towards it, but remaining under the x-axis for large negative \(x\). For \(x = 0\), calculate \(f(0) = -1 + 4 = 3\), and plot this point as it is above the x-axis.

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Key Concepts

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

Understanding Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A general form of an exponential function is given by \(a^\{x\}\), where \'a\' is a positive real number, and \'x\' is the variable. Depending on the base, an exponential function can either increase or decrease:
  • If \(a > 1\), the function exponentially increases.
  • If \(0 < a < 1\), the function exponentially decreases.
In the exercise, the base of the function is \(\frac{1}{2}\), which is less than 1, meaning it represents a decaying or decreasing exponential function. Such functions are essential in modeling processes like radioactive decay or depreciation, where quantities decrease over time at a rapidly changing rate.
Exploring Graph Transformations
Graph transformations are methods used to alter the appearance of a graph without changing its core properties. These transformations include changes in position, orientation, and size.
Basic transformations can be summarized as follows:
  • Shifts: Move the graph horizontally or vertically.
  • Stretching and Compressing: Change the width or height of the graph.
  • Reflections: Flip the graph across an axis.
In our exercise, the graph of \(g(x) = \left(\frac{1}{2}\right)^x\) was modified to form \(f(x) = -\left(\frac{1}{2}\right)^x + 4\) through a series of transformations. Identifying these changes helps in sketching the graph accurately.
Reflection Across the X-Axis
Reflection involves flipping a graph over a specific axis, creating a mirror image. When a function is multiplied by \(-1\), it flips across the x-axis.
In the example from the exercise, the exponential function \(g(x) = \left(\frac{1}{2}\right)^x\) becomes \(-\left(\frac{1}{2}\right)^x\). This reflection inverts the graph, causing points that were originally above the x-axis to now appear below it. Despite changing its position, the reflected graph retains its decreasing nature.
Reflections are a critical tool in graph transformations, providing a straightforward way to change the direction of a function's growth or decay while preserving its shape.
Understanding Vertical Shifts
Vertical shifts involve moving the entire graph up or down along the y-axis. This transformation is achieved by adding or subtracting a constant to the function.
For the function \(f(x) = -\left(\frac{1}{2}\right)^x + 4\), the \(+4\) causes a vertical shift upwards by 4 units. This shift alters the graph's asymptote from \(y = 0\) to \(y = 4\).
Vertical shifts do not change the shape of the graph but redefine its position along the y-axis. These shifts are essential for adjusting the baseline level in various applications, such as accounting for an initial quantity in scientific models.

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Most popular questions from this chapter

The population \(N(t)\) (in millions) of the United States \(t\) years after 1980 may be approximated by the formula \(N(t)=231 e^{11,013t}.\) When will the population be twice what it was in \(1980?\)

Approximate \(x\) to three significant figures. (a) \(\log x=1.8965\) (b) \(\log x=4.9680\) (c) \(\log x=-2.2118\) (d) \(\ln x=3.7\) (e) \(\ln x=0.95\) (f) \(\ln x=-5\)

government receipts (in billions of dollars) for selected years are listed in the table. $$\begin{array}{|lcccc|}\hline \text { Year } & 1910 & 1930 & 1950 & 1970 \\\\\hline \text { Recelpts } & 0.7 & 4.1 & 39.4 & 192.8 \\\\\hline \text { Year } & 1980 & 1990 & 2000 \\\\\hline \text { Receipts }& 517.1 & 1032.0 & 2025.2 \\\\\hline\end{array}$$ (a) Let \(x=0\) correspond to the year \(1910 .\) Plot the data, together with the functions \(f\) and \(g\) : (1) \(f(x)=0.786(1.094)^{x}\) (2) \(g(x)=0.503 x^{2}-27.3 x+149.2\) (b) Determine whether the exponential or quadratic function better models the data. (c) Use your choice in part (b) to graphically estimate the year in which the federal government first collected S1 trillion.

For manufacturers of computer chips, it is important to consider the fraction \(F\) of chips that will fail after \(t\) years of service. This fraction can sometimes be approximated by the formula \(F=1-e^{-c t},\) where \(c\) is a positive constant. (a) How does the value of \(c\) affect the reliability of a chip? (b) If \(c=0.125,\) after how many years will 35 \(\%\) of the chips have failed?

In statistics, the probability density function for the normal distribution is defined by $$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-z^{2 / 2}} \quad \text { with } \quad z=\frac{x-\mu}{\sigma}$$ where \(\mu\) and \(\sigma\) are real numbers ( \(\mu\) is the mean and \(\sigma^{2}\) is the variance of the distribution). Sketch the graph of \(f\) for the case \(\sigma=1\) and \(\mu=0\).
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