The x- and y- intercepts for the equation y = 9 − x 2 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 66E
To determine

To calculate: The x- and y- intercepts for the equation y=9−x2 and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Expert Solution

The x-interceptsare ±3 and y-intercept is 9 . The equation is symmetric about y-axis. Graph of the equation is provided below,

Explanation of Solution

Given information:

The equation y=9x2 .

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 y=9x2 . Construct a table to evaluate the value of y for different values of x.

Substitute the point x=0 in the equation y=9x2 ,

y=902y=9

Substitute the point x=1 in the equation y=9x2 ,

y=912y=8

Substitute the point x=2 in the equation y=9x2 ,

y=922y=5

Substitute the point x=3 in the equation y=9x2 ,

y=932y=0

Construct a table with the values obtained above,

xy(x,y)09(0,9)18(1,8)25(2,5)30(3,0)

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

The graph of the equation is provided below y=9x2 .

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 y=9x2 ,

0=9x2x2=9x=±3

Therefore, x-intercepts are ±3 .

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 y=9x2 ,

y=902y=9

Therefore, y-intercept is 9 .

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 y=9x2 ,

(y)=9x2y=9x2

The equation is changed. Therefore, the equation y=9x2 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 y=9x2 ,

y=9(x)2y=9x2

The equation is unchanged. Therefore, the equation y=9x2 is 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 y=9x2 ,

y=9(x)2y=9x2

The equation is changed. Therefore, the equation y=9x2 is not symmetricabout the origin.

Thus, the x-intercepts are ±3 and y-intercept is 9 . The equation is symmetric about y-axis.Graph of the equation y=9x2 is provided below,

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