The x- and y- intercepts for the equation 8 y = x 3 and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.8, Problem 64E
To determine

To calculate: The x- and y- intercepts for the equation 8y=x3 and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Expert Solution

The x-intercept is 0 and y-intercept is 0 . The equation is symmetric about origin.Graph of the equation 8y=x3 is provided below,

Explanation of Solution

Given information:

The equation 8y=x3 .

Formula used:

The function is symmetric about the x-axis, when y is replaced by y , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by x , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by y and x is replaced by x , the equation remains unchanged.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Calculation:

It is provided that the equation is 8y=x3 . Construct a table to evaluate the value of y for different values of x.

Substitute the point x=0 in the equation 8y=x3 ,

8y=03y=0

Substitute the point x=1 in the equation 8y=x3 ,

8y=13y=18

Substitute the point x=2 in the equation 8y=x3 ,

8y=23y=88y=1

Substitute the point x=4 in the equation 8y=x3 ,

8y=43y=648y=8

Construct a table with the values obtained above,

xy(x,y)00(0,0)118(1,18)21(2,1)48(4,8)

In the coordinate plane plot the points obtained above and connect them through a line.

The graph of the equation is provided below 8y=x3 .

Recall that the x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

Substitute y=0 in the equation 8y=x3 ,

8(0)=x3x=0

Therefore, x-intercepts is 0.

Recall that the y-intercepts are the points on x-axis where the graph of the equation intersects the y-axis.

Substitute x=0 in the equation 8y=x3 ,

8y=03y=0

Therefore, y-intercept is 0 .

Recall that the function is symmetric about the x-axis, when y is replaced by y , the equation remains unchanged.

Replace y by y in the equation 8y=x3 ,

8(y)=x38y=x3

The equation is changed. Therefore, the equation 8y=x3 is not symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by x , the equation remains unchanged.

Replace x by x in the equation 8y=x3 ,

8y=(x)38y=x3

The equation is changed. Therefore, the equation 8y=x3 is not symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by y and x is replaced by x , the equation remains unchanged.

Replace x by x and y by y in the equation 8y=x3 ,

8y=(x)38y=x38y=x3

The equation is unchanged. Therefore, the equation 8y=x3 is symmetricabout the origin.

Thus, the x-intercept is 0 and y-intercept is 0 . The equation is symmetric about origin.Graph of the equation 8y=x3 is provided below,

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