Create a table of values and remember that sqrt(x) only equals an integer for perfect cubes, x^3=x* x* x.
Description: See solution. Graph:
Practice makes perfect
To graph the function y=sqrt(x)-2, we can create a table of values. Since the function takes the cube root of x, to make things easier, we will choose values for x that are perfect cubes. Remember that a perfect cube is any integer y that can be written as the product of an integer x three times.
y=x^3=x* x* x
Cubing the numbers from 1 to 4, we have the following.
1^3& =1 * 1 *1 =1
2^3& =2 * 2 *2 =8
3^3& =3 * 3 *3 =27
4^3& =4 * 4 *4 =64
We can use the positives and negatives of these values to create the table.
|c|c|c|c|
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x & sqrt(x)-2 & Write as a Cube & y [0.5em]
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-64 & sqrt(-64)-2 & sqrt((-4)^3)-2 & - 6 [0.5em]
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-27 & sqrt(-27)-2 & sqrt((-3)^3)-2 & - 5 [0.5em]
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-8 & sqrt(-8)-2 & sqrt((-2)^3)-2 & - 4 [0.5em]
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-1 & sqrt(-1)-2 & sqrt((-1)^3)-2 & - 3 [0.5em]
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0 & sqrt(0)-2 & sqrt(0^3)-2 & - 2 [0.5em]
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1 & sqrt(1)-2 & sqrt(1^3)-2 & - 1 [0.5em]
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8 & sqrt(8)-2 & sqrt(2^3)-2 & 0 [0.5em]
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27 & sqrt(27)-2 & sqrt(3^3)-2 & 1 [0.5em]
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64 & sqrt(64)-2 & sqrt(4^3)-2 & 2 [0.5em]
We will graph the function by plotting the (x,y) points from the table.
To describe the graph above, we can use the description guidelines from the previous exercise.
Shape: The shape of the graph does not show a line.
Rate of change: The shape of the graph and the increasing distance between points prove that the rate of change is not constant.
Symmetry: The graph looks symmetrical about the y-intercept. If we were to rotate the positive portion 180^(∘) clockwise, it would lie on top of the negative section.
Possible x- and y-values: Since the graph extends infinitely in all directions, x and y can both take on all real numbers.
Starting/stopping point: There is no stopping or starting point, because the graph extends infinitely.
Minimum/Maximum: There is no minimum or maximum point, because the graph extends infinitely.
