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4.3: Exercises

Last updated

May 2, 2022
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- 4.2: Finding zeros, maxima, and minima
- 5: Basic Functions and Transformations

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 equation in the standard window.

$y=3x-5$
$y=x^2-3x-2$
$y= x^4-3x^3+2x-1$
$y=\sqrt{x^2-4}$
$y=\dfrac 1 {x+2}$
$y=|x+3|$

For the last exercise, the absolute value is obtained by pressing $\boxed {\text{math}}\boxed {\triangleright } \boxed {\text{enter}}$ .

Answer

$clipboard_ec28580a0cb475221ae74f65a7b875a56.png$
$clipboard_e9639052f0518480c0a6772d864862076.png$
$clipboard_ea1d7f62ea6557b37425bd10c02a0af69.png$
$clipboard_e315786a025fabfb35de4054820a7eb18.png$
$clipboard_e1ec1c2b8561b7e8da1309cb0bed4db69.png$
$clipboard_e6d7f2f5f1bd0d3db032f5a0fb5a50e62.png$

Exercise $\PageIndex{2}$

Solve the equation for $y$ and graph all branches in the same window.

$x^2+y^2=4$
$(x+5)^2+y^2=15$
$(x-1)^2+(y-2)^2 = 9$
$y^2+x^2-8x-14=0$
$y^2=x^2+3$
$y^2=-x^2+77$

Answer

$y_{1}=\sqrt{4-x^{2}}$ , $y_{2}=-\sqrt{4-x^{2}}$ , $clipboard_e555203ea583554191525d0fc203c89b5.png$
$y_{1}=\sqrt{15-(x+5)^{2}}$ , $y_{2}=-\sqrt{15-(x+5)^{2}}$ , $clipboard_e036464d3ffa47eb8f571fee5ffbbeeba.png$
$y_1=2+\sqrt{9-(x-1)^{2}}$ , $y_{2}=2-\sqrt{9-(x-1)^{2}}$ , $clipboard_e0158735c0a9f6b3678a74ddb74b6c1ef.png$
$y_{1}=\sqrt{-x^{2}+8 x+14}$ , $y_2=-\sqrt{-x^{2}+8 x+14}$ , $clipboard_e4d4e8cce97279d973b5dc08559a13d96.png$
$y_{1}=\sqrt{x^{2}+3}$ , $y_{2}=-\sqrt{x^{2}+3}$ , $clipboard_e5e95ef043f4e9dc013b8e7d9a601f39e.png$
$y_{1}=\sqrt{-x^{2}+77}$ , $y_{2}=-\sqrt{-x^{2}+77}$ , $clipboard_e3af58a942e94955990b7dad10906a5f5.png$

Exercise $\PageIndex{3}$

Find all zeros of the given function. Round your answer to the nearest hundredth.

$f(x)=x^2+3x+1$
$f(x)=x^4-3x^2+2$
$f(x)=x^3+2x-6$
$f(x)=x^5-11x^4+20x^3+88x^2-224x+1$
$f(x)=x^3-5x^2+2x+3$
$f(x)=\sqrt{2^x-3}-2x+3$
$f(x)=0.04 x^3-0.02x^2-0.5174x+0.81$
$f(x)=0.04 x^3-0.02x^2-0.5175x+0.81$
$f(x)=0.04 x^3-0.02x^2-0.5176x+0.81$

Answer

$x \approx-2.62, x \approx-0.38$
$x=\pm 1, x=\pm \sqrt{2} \approx \pm 1.41$
$x \approx 1.46$
$x \approx-2.83, x \approx 0.01, x \approx 2.82, x \approx 4.01, x \approx 7.00$
$x \approx-0.578, x \approx 1.187, x \approx 4.388$
$x \approx 1.61, x=2, x \approx 6.91$
$x \approx-4.00$
$x=-4, x=2.25$
$x \approx-4.00, x \approx 2.22, x \approx 2.28$

Exercise $\PageIndex{4}$

Find all solutions of the equation. Round your answer to the nearest thousandth.

$x^3+3=x^5+7$
$4x^3+6x^2-3x-2=0$
$\dfrac{2x}{x-3}=\dfrac{x^2+2}{x+1}$
$5^{3x+1}=x^5+6$
$x^3+x^2=x^4-x^2+x$
$3x^2=x^3-x^2+3x$

Answer

$x \approx-1.488$
$x \approx-1.764, x \approx-0.416, x \approx 0.681$
$x \approx 5.220$
$x \approx-1.431, x \approx 0.038$
$x \approx-1.247, x=0, x \approx 0.445, x \approx 1.802$
$x=0, x=1, x=3$

Exercise $\PageIndex{5}$

Graph the equation. Determine how many maxima and minima the graph has. To this end, resize the graphing window (via the zoom-in, zoom-out, and zoom-box functions of the calculator) to zoom into the maxima or minima of the graph.

$y=x^2-4x+13$
$y=-x^2+x-20$
$y=2x^3 -5x^2+3x$
$y=x^4-5x^3+8x^2-5x+1$

Answer

There is one minimum. Zoom out for the graph. $clipboard_ea89937a4d3facf7eac223808ff617276.png$
There is one maximum. Resize the window to Ymin= −100. $clipboard_e3a05feccdf66f88d72b6bb4706e4b099.png$
There is one local maximum and one local minimum. The graph with Xmin=−4, Xmax= 4, Ymin= −2, Ymax= 2 is below. $clipboard_e89ccc66825465c44830ab3e3a4ec160b.png$
Zooming into the graph reveals two local minima and one local maximum. We graph the function with Xmin=−2, Xmax= 4, Ymin= −1.3, Ymax= 0.5. $clipboard_e679223dd8983c143a017fe9d69087df9.png$

Exercise $\PageIndex{6}$

Approximate the (local) maxima and minima of the graph. Round your answer to the nearest tenth.

$y=x^3+2x^2-x+1$
$y=x^3-5x^2+8x-3$
$y=-x^4+3x^3+x^2+2$
$y=x^4-x^3-4x^2+6x+2$
$y=x^4-x^3-4x^2+8x+2$
$y=x^4-x^3-4x^2+7x+2$

Answer

$clipboard_e29efa53019b5b1a94e684e58e5a6141f.png$
$clipboard_ed9a91c9ecd139448b4f5e54556561d68.png$ $clipboard_e82043561f8991ff0f6c44a6fe382b8a9.png$
$clipboard_e9923bffd128206319d4a7da325631da7.png$ $clipboard_e1c88d6a252006a41c1b172532849a6a1.png$
$clipboard_e1077580760809416dee5913010b0f7da.png$ $clipboard_e39115538cd162dcd08681b7867facf4b.png$
$clipboard_eab01b98f8b96d5ca97de38f9ed227764.png$ (there is only one minimum and no maximum in part (e))