4.3: Exercises

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 \(\boxed {\text{math}}\boxed {\triangleright } \boxed {\text{enter}}\).

Answer

1. 
2. 
3. 
4. 
5. 
6. 

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. \(y^2=-x^2+77\)

Answer

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. \(f(x)=0.04 x^3-0.02x^2-0.5176x+0.81\)

Answer

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. \(3x^2=x^3-x^2+3x\)

Answer

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. \(y=x^4-5x^3+8x^2-5x+1\)

Answer

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. \(y=x^4-x^3-4x^2+7x+2\)

Answer

1. 
2. 
3. 
4. 
5. (there is only one minimum and no maximum in part (e))