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Chapter 13: Problem 35

Use a calculator to graph the function \(y=2^{|x|}\).

Short Answer

Expert verified
The graph of \(y = 2^{|x|}\) is a V-shaped curve, symmetric around the y-axis, with exponential arms extending away from it.

Step by step solution

01

Understand the Function

The given function is \(y = 2^{|x|}\). This is an exponential function where the base is 2, and the exponent is the absolute value of \(x\). Since the exponent is always non-negative due to the absolute value, the function will always produce positive y-values.
02

Determine Key Points

Calculate some key points to understand the behavior of the graph. Evaluate the function at \(-2, -1, 0, 1,\) and \(2\):1. \(x = -2\), \(y = 2^{|-2|} = 2^2 = 4\)2. \(x = -1\), \(y = 2^{|-1|} = 2^1 = 2\)3. \(x = 0\), \(y = 2^{|0|} = 2^0 = 1\)4. \(x = 1\), \(y = 2^{|1|} = 2^1 = 2\)5. \(x = 2\), \(y = 2^{|2|} = 2^2 = 4\)
03

Analyze Symmetry

The function \(y = 2^{|x|}\) is symmetric with respect to the y-axis. This is because \(|x| = |-x|\), which results in the same y-value for \(x\) and \(-x\).
04

Graph the Function

Using a graphing calculator, plot the points \((-2, 4)\), \((-1, 2)\), \((0, 1)\), \((1, 2)\), and \((2, 4)\). Connect these points with a smooth curve to form a V-shaped graph that increases exponentially away from the y-axis. The left side of the graph mirrors the right due to symmetry.

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the function \(y = 2^{|x|}\), the base is 2. Exponential growth occurs as the absolute value of \(x\) increases, leading to larger results for \(y\).

Here, the exponent is \(|x|\), making it particularly interesting since absolute values always yield non-negative numbers. This ensures \(y\) is always positive. Such a function grows rapidly as \(|x|\) increases, demonstrating exponential growth in both positive and negative directions of \(x\).

To put it simply:
  • Base: Constant (2 in this case)
  • Exponent: Variable (here, the absolute value of \(x\))
  • Output: Always positive, increases exponentially
Absolute Value
The absolute value function, denoted as \(|x|\), always returns the non-negative version of \(x\). This means whether \(x\) is positive or negative, \(|x|\) is the non-negative counterpart. In our function \(y = 2^{|x|}\), the absolute value significantly influences the outcome of the function.

For instance, \(2^{|x|}\) means whether \(x\) is 2 or -2, \(|x| = 2\). This feature simplifies understanding; you don't need separate calculations for negative and positive values of \(x\).

Remember:
  • \(|x| = x\) if \(x \geq 0\)
  • \(|x| = -x\) if \(x < 0\)
  • This ensures the output of \(y = 2^{|x|}\) remains consistent in all cases.
Symmetry in Graphs
Symmetry in graphs often simplifies understanding and plotting. For the function \(y = 2^{|x|}\), there is symmetry about the y-axis. This is because \(|x|\) produces identical results for both positive \(x\) and negative \(x\).

This particular symmetry tells us:
  • The right side of the graph mirrors the left side.
  • You only need to evaluate the function in one half of the plane to understand its full behavior.
  • This V-shape resulting from the symmetry is a hallmark of functions involving absolute values.
In practical terms, symmetry can help in anticipating values without needing repetitive calculations.
Graphing Calculator Usage
Using a graphing calculator can significantly aid in visualizing functions, especially ones as dynamic as exponential functions. Graphing \(y = 2^{|x|}\) can initially seem tricky due to the absolute values and exponential growth, but a calculator streamlines understanding.

Here's a simple process:
  • Enter the function into the graphing calculator: Most calculators have a function mode where you can input \(y = 2^{|x|}\).
  • Set a suitable range: Start with a small window for \(x\) and \(y\) to see the central part of the graph before expanding it.
  • Analyze symmetry: Notice how both halves of the graph mirror across the y-axis, reflecting the principles of symmetry.
  • Use trace features: Many calculators allow you to trace the graph to find exact points easily, reinforcing the plotted values.
By utilizing these tools, you can efficiently comprehend and display even complex graphs.

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Most popular questions from this chapter

Plot the indicated semilogarithmic graphs for the following application. In a particular electric circuit, called a low-pass filter, the input voltage \(V_{i}\) is across a resistor and a capacitor, and the output voltage \(V_{0}\) is across the capacitor (see Fig. 13.28 ). The voltage gain \(G\) (in \(d B\) ) is given by $$G=20 \log \frac{1}{\sqrt{1+(\omega T)^{2}}}$$ where \(\tan \phi=-\omega T\) Here, \(\phi\) is the phase angle of \(V_{0} / V_{i .}\) For values of \(\omega T\) of 0.01,0.1,0.3 \(1.0,3.0,10.0,30.0,\) and \(100,\) plot the indicated graphs. These graphs are called a Bode diagram for the circuit. Calculate values of \(G\) for the given values of \(\omega T\) and plot a semilogarithmic graph of \(G\) vs. \(\omega T\)

\(\text {Solve the given problems.}\) When a person ingests a medication capsule, it is found that the rate \(R\) (in \(\mathrm{mg} / \mathrm{min}\) ) that it enters the bloodstream in time \(t\) (in \(\min\) ) is given by \(\log _{10} R-\log _{10} 5=t \log _{10} 0.95 .\) Solve for \(R\) as a function of \(t\)

Use a calculator to solve the given equations. $$\log 4 x+\log x=2$$

Perform the indicated operations. The velocity \(v\) of a rocket when its fuel is completely burned is given by \(v=u \log _{e}\left(w_{0} / w\right),\) where \(u\) is the exhaust velocity, \(w_{0}\) is the liftoff weight, and \(w\) is the burnout weight. Solve for \(w\)

\(\text {Solve the given problems.}\) In analyzing the power gain in a microprocessor circuit, the equation \(N=10\left(2 \log _{10} I_{1}-2 \log _{10} I_{2}+\log _{10} R_{1}-\log _{10} R_{2}\right)\) is used. Express this with a single logarithm on the right side.
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