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13.3: Exercises

Last updated

May 2, 2022
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- 13.2: Logarithmic functions and their graphs
- 14: Properties of Exponentials and Logarithms

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Graph the following functions with the calculator.

$y=5^x$
$y=1.01^x$
$y=\left (\dfrac 1 3 \right )^x$
$y=0.97^x$
$y=3^{-x}$
$y=\left (\dfrac 1 3 \right )^{-x}$
$y=e^{x^2}$
$y=0.01^x$
$y=1^x$
$y=e^{x}+1$
$y=\dfrac{e^x-e^{-x}}{2}$
$y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$

The last two functions are known as the hyperbolic sine, $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ , and the hyperbolic tangent, $\tanh(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$ . Recall that the hyperbolic cosine $\cosh(x)=\dfrac{e^x+e^{-x}}{2}$ was already graphed in example hyperbolic-cosine.

Answer

$clipboard_e929886fc88b1e5ada4f15b02d6eec21a.png$
$clipboard_eaf306e99bf7fb70aedacd464ecc7821a.png$
$clipboard_eee8ff32d64022afbc5328894d6103f6c.png$
$clipboard_eaf1f5a68c1e3100f5058af3da8458372.png$
same as c) since $y=\left(\dfrac{1}{3}\right)^{x}=3^{-x}$
$clipboard_ee2aa51fd0c9bf3df80bde31357a9ddde.png$
$clipboard_e92023a7d7ae66b34e4a1bd81dd99fdd4.png$
$clipboard_e8bf1d20fe2f629b9634fb50fc7352e2f.png$
$clipboard_e9802b80547a2ceb8b2286b5a5ce7215d.png$
$clipboard_e4cb72ca83788db3e3c3fa3027a69d5bd.png$
$clipboard_e8a306ec0254118b085cb453931a64337.png$
$clipboard_e6c76d1311825ca28327e82b37ea04265.png$

Exercise $\PageIndex{2}$

Graph the given function. Describe how the graph is obtained by a transformation from the graph of an exponential function $y=b^x$ (for appropriate base $b$ ).

$y=0.1\cdot 4^x$
$y=3\cdot 2^x$
$y=(-1)\cdot 2^x$
$y=0.006\cdot 2^x$
$y=e^{-x}$
$y=e^{-x}+1$
$y=(\dfrac 1 2)^{x}+3$
$y=2^{x-4}$
$y=2^{x+1}-6$

Answer

$y=4^{x}$ is compressed towards the x-axis by the factor $0.1$ $clipboard_e9a8a160a6cac2c21b6f0059ae20bcd1d.png$
$y = 2^x$ stretched away from $x$ -axis $clipboard_eafcdca9fe269fe2e98bc970f88bfb13d.png$
$y = 2^x$ reflected about the $x$ -axis $clipboard_e64e2e9be6cc3d7423cbe61a6b069fffb.png$
$y = 2^x$ compressed towards the $x$ -axis $clipboard_ecb09bb72f8740e100c7b907d68685241.png$
$y = e^x$ reflected about the $y$ -axis $clipboard_e1c93f931cc705f63520aebee0fa72207.png$
$y = e^x$ reflected about the $y$ -axis and shifted up by $1$ $clipboard_e11c91472da5fb45476dcba4a9321a715.png$
$y=\left(\dfrac{1}{2}\right)^{x}$ shifted up by $3$ $clipboard_ec8ff260d6a742fe9db9b1667a93994eb.png$
$y = 2^x$ shifted to the right by $4$ $clipboard_e442de036592464da648a4aa983f3c5fb.png$
$y = 2^x$ shifted to the left by $1$ and down by $6$ $clipboard_e530960e6d6e975674b92a314588f358a.png$