BuyFind

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 1.8, Problem 71E
To determine

To calculate: The x- and y- intercepts for the equation x+y2=4 and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Expert Solution

Answer to Problem 71E

The x-intercept is 4 and y-intercepts are ±2 . The equation is symmetric about x-axis.Graph of the equation x+y2=4 is provided below,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 71E , additional homework tip  1

Explanation of Solution

Given information:

The equation x+y2=4 .

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 x+y2=4 . Construct a table to evaluate the value of y for different values of x.

Substitute the point x=0 in the equation x+y2=4 ,

  0+y2=4y2=4y=±2

Substitute the point x=3 in the equation x+y2=4 ,

  3+y2=4y2=1y=±1

Construct a table with the values obtained above,

  xy(x,y)02(0,2)02(0,2)31(3,1)31(3,1)

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

The graph of the equation is provided below x+y2=4 .

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 71E , additional homework tip  2

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 x+y2=4 ,

  x+02=4x=4

Therefore, x-intercept is 4.

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 x+y2=4 ,

  0+y2=4y2=4y=±2

Therefore, y-intercept are ±2 .

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 x+y2=4 ,

  x+(y)2=4x+y2=4

The equation is unchanged. Therefore, the equation x+y2=4 is 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 x+y2=4 ,

  x+y2=4

The equation is changed. Therefore, the equation x+y2=4 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 x+y2=4 ,

  x+(y)2=4x+y2=4

The equation is changed. Therefore, the equation x+y2=4 is not symmetricabout the origin.

Thus, the x-intercept is 4 and y-interceptsare ±2 . The equation is symmetric about y-axis.Graph of the equation x+y2=4 is provided below,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 71E , additional homework tip  3

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