Chapter T, Problem 5CDT

Without using a calculator, make a rough sketch of the graph.

1. (a) y = x³
2. (b) y = (x + 1)³
3. (c) y = (x − 2)³ + 3
4. (d) y = 4 − x²
5. (e) $y=\sqrt{x}$
6. (f) $y=2\sqrt{x}$
7. (g) y = −2^x
8. (h) y = 1 + x⁻¹

To determine

To sketch: The graph of the function $y=x^3$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=x^3$.

x	$-2$	$-1$	0	$1$	$2$
y	$-8$	$-1$	0	1	8

The graph for the given function $y=x^3$ is shown below in Figure 1.

From Figure 1 it is observed that, the graph of the function $y=x^3$ is unbounded region.

To determine

To sketch: The graph of the function $y={\left(x+1\right)}^3$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y={\left(x+1\right)}^3$.

x	$-2$	$-1$	0	$1$	$2$
y	$-1$	0	1	8	27

The graph for the given function $y={\left(x+1\right)}^3$ is shown below in Figure 2.

From Figure 2 it is observed that, the graph of the function $y={\left(x+1\right)}^3$ is unbounded region. The graph of the function $y={\left(x+1\right)}^3$ is translated right by 1 units.

To determine

To sketch: The graph of the function $y={\left(x-2\right)}^3+3$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y={\left(x-2\right)}^3$.

x	$-2$	$-1$	0	$1$	$2$
y	$-64$	$-27$	$-8$	$-1$	0

The graph for the given function $y={\left(x-2\right)}^3$ is shown below in Figure 3.

According to the transaltion and reflection of the function the graph of the function $y={\left(x-2\right)}^3$ is obtained if the graph of $y={\left(x-2\right)}^3$ is translated upward by 3 units.

The graph for the given function $y={\left(x-2\right)}^3+3$ is shown below in Figure 4.

From Figure 2 , it is observed that the graph of the function $y={\left(x-2\right)}^3+3$ is obtained if the graph of the function $y={\left(x-2\right)}^3+3$ is translated right by 2 units and upward by 3 units.

To determine

To sketch: The graph of the function $y=4-x^2$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=x^2$.

x	$-2$	$-1$	0	$1$	$2$
y	4	1	0	1	4

The graph for the given function $y=x^2$ is shown below in Figure 5.

According to the transaltion and reflection of the function the graph of the function $y=x^2$ is obtained if the graph of $y=x^2$ is translated upward by $-4$ units.

The graph for the given function $y=4-x^2$ is shown below in Figure 6.

From Figure 6 it is observed that, the graph of the function $y=4-x^2$ is unbounded region.

To determine

To sketch: The graph of the function $y=\sqrt{x}$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=\sqrt{x}$.

x	1	2	3	4	5
y	1	$\sqrt{2}$	$\sqrt{3}$	2	$\sqrt{5}$

The graph for the given function $y=\sqrt{x}$ is shown below in Figure 7.

From Figure 7 it is observed that, the graph of the function $y=\sqrt{x}$ is unbounded region.

To determine

To sketch: The graph of the function $y=2\sqrt{x}$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=2\sqrt{x}$.

x	1	2	3	4	5
y	2	$2\sqrt{2}$	$2\sqrt{3}$	4	$2\sqrt{5}$

The graph for the given function $y=2\sqrt{x}$ is shown below in Figure 8.

From Figure 8 it is observed that, the graph of the function $y=2\sqrt{x}$ is unbounded region.

To determine

To sketch: The graph of the function $y=-2^x$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=2^x$.

x	$-1$	0	1	2	3
y	$\frac{1}{2}$	1	2	4	8

The graph for the given function $y=2^x$ is shown below in Figure 9.

Reflect the graph of $y=2^x$ about the x axis then obtained $y=-2^x$. The graph for the given function $y=-2^x$ is shown below in Figure 10.

From Figure 10 it is observed that, the graph of the function $y=-2^x$ is unbounded region.

To determine

To sketch: The graph of the function $y=1+x^{-1}$ without using a calculator.

Explanation of Solution

Compute a table for various values of x and then use the value of the function to sketch a smooth curve of $y=x^{-1}$.

x	$-\frac{1}{2}$	1	$\frac{1}{2}$	$-\frac{1}{3}$	$-\frac{1}{4}$
y	$-2$	1	2	$-3$	$-4$

The graph for the given function $y=x^{-1}$ is shown below in Figure 11.

According to the translation of the function the graph of the function $y=x^{-1}$ is obtained if the graph of $y=x^{-1}$ is translated upward by 1 units.

The graph for the given function $y=1+x^{-1}$ is shown below in Figure 12.

From Figure 12 it is observed that, the graph of the function $y=1+x^{-1}$ is unbounded region.