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 62E
To determine

To calculate: The x- and y- intercepts for the equation y=x2+2 and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Expert Solution

Answer to Problem 62E

There arenox-intercepts and y-intercept is 2 . The equation is symmetric about y-axis.Graph of the equation y=x2+2 is provided below,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 62E , additional homework tip  1

Explanation of Solution

Given information:

The equation y=x2+2 .

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=x2+2 . Construct a table to evaluate the value of y for different values of x.

Substitute the point x=0 in the equation y=x2+2 ,

  y=02+2y=2

Substitute the point x=1 in the equation y=x2+2 ,

  y=12+2y=3

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

  y=22+2y=4+2y=6

Substitute the point x=3 in the equation y=x2+2 ,

  y=32+2y=11

Construct a table with the values obtained above,

  xy(x,y)02(0,2)13(1,3)26(2,6)311(3,11)

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

The graph of the equation is provided below y=x2+2 .

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 62E , 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 y=x2+2 ,

  0=x2+2x2=2

Therefore, nox-intercepts are there.

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=x2+2 ,

  y=02+2y=2

Therefore, y-intercept is 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 y=x2+2 ,

  y=x2+2

The equation is changed. Therefore, the equation y=x2+2 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=x2+2 ,

  y=(x)2+2y=x2+2

The equation is unchanged. Therefore, the equation y=x2+2 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=x2+2 ,

  y=(x)2+2y=x2+2

The equation is changed. Therefore, the equation y=x2+2 is not symmetricabout the origin.

Thus, there are nox-intercepts and y-intercept is 2 . The equation is symmetric about y-axis.Graph of the equation y=x2+2 is provided below,

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 62E , additional homework tip  3

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