# 4.3: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Exercise $$\PageIndex{1}$$

Graph the equation in the standard window.

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

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

## Exercise $$\PageIndex{2}$$

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

1. $$x^2+y^2=4$$
2. $$(x+5)^2+y^2=15$$
3. $$(x-1)^2+(y-2)^2 = 9$$
4. $$y^2+x^2-8x-14=0$$
5. $$y^2=x^2+3$$
6. $$y^2=-x^2+77$$
1. $$y_{1}=\sqrt{4-x^{2}}$$, $$y_{2}=-\sqrt{4-x^{2}}$$,
2. $$y_{1}=\sqrt{15-(x+5)^{2}}$$, $$y_{2}=-\sqrt{15-(x+5)^{2}}$$,
3. $$y_1=2+\sqrt{9-(x-1)^{2}}$$, $$y_{2}=2-\sqrt{9-(x-1)^{2}}$$,
4. $$y_{1}=\sqrt{-x^{2}+8 x+14}$$, $$y_2=-\sqrt{-x^{2}+8 x+14}$$,
5. $$y_{1}=\sqrt{x^{2}+3}$$, $$y_{2}=-\sqrt{x^{2}+3}$$,
6. $$y_{1}=\sqrt{-x^{2}+77}$$, $$y_{2}=-\sqrt{-x^{2}+77}$$,

## Exercise $$\PageIndex{3}$$

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

1. $$f(x)=x^2+3x+1$$
2. $$f(x)=x^4-3x^2+2$$
3. $$f(x)=x^3+2x-6$$
4. $$f(x)=x^5-11x^4+20x^3+88x^2-224x+1$$
5. $$f(x)=x^3-5x^2+2x+3$$
6. $$f(x)=\sqrt{2^x-3}-2x+3$$
7. $$f(x)=0.04 x^3-0.02x^2-0.5174x+0.81$$
8. $$f(x)=0.04 x^3-0.02x^2-0.5175x+0.81$$
9. $$f(x)=0.04 x^3-0.02x^2-0.5176x+0.81$$
1. $$x \approx-2.62, x \approx-0.38$$
2. $$x=\pm 1, x=\pm \sqrt{2} \approx \pm 1.41$$
3. $$x \approx 1.46$$
4. $$x \approx-2.83, x \approx 0.01, x \approx 2.82, x \approx 4.01, x \approx 7.00$$
5. $$x \approx-0.578, x \approx 1.187, x \approx 4.388$$
6. $$x \approx 1.61, x=2, x \approx 6.91$$
7. $$x \approx-4.00$$
8. $$x=-4, x=2.25$$
9. $$x \approx-4.00, x \approx 2.22, x \approx 2.28$$

## Exercise $$\PageIndex{4}$$

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

1. $$x^3+3=x^5+7$$
2. $$4x^3+6x^2-3x-2=0$$
3. $$\dfrac{2x}{x-3}=\dfrac{x^2+2}{x+1}$$
4. $$5^{3x+1}=x^5+6$$
5. $$x^3+x^2=x^4-x^2+x$$
6. $$3x^2=x^3-x^2+3x$$
1. $$x \approx-1.488$$
2. $$x \approx-1.764, x \approx-0.416, x \approx 0.681$$
3. $$x \approx 5.220$$
4. $$x \approx-1.431, x \approx 0.038$$
5. $$x \approx-1.247, x=0, x \approx 0.445, x \approx 1.802$$
6. $$x=0, x=1, x=3$$

## 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.

1. $$y=x^2-4x+13$$
2. $$y=-x^2+x-20$$
3. $$y=2x^3 -5x^2+3x$$
4. $$y=x^4-5x^3+8x^2-5x+1$$
1. There is one minimum. Zoom out for the graph.
2. There is one maximum. Resize the window to Ymin= −100.
3. There is one local maximum and one local minimum. The graph with Xmin=−4, Xmax= 4, Ymin= −2, Ymax= 2 is below.
4. 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.

## Exercise $$\PageIndex{6}$$

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

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