Symmetry and Function Behavior

The video tutorial explains how to test for symmetry in equations around the x-axis, y-axis, and origin. It provides step-by-step instructions for each type of symmetry using different equations as examples. The process involves substituting variables and simplifying equations to determine if they match the original, indicating symmetry.

What is the first step in testing for symmetry in equations?

• Replace all x's with y's

• Replace all variables with zero

• Replace all y's with negative y's

• Replace all numbers with their negatives

When testing for symmetry around the x-axis, what should you do after replacing y with -y?

• Replace x with -x

• Multiply the equation by 2

• Check if the equation is linear

• Simplify the equation

What indicates that an equation is symmetric around the x-axis?

• The simplified equation is different from the original

• The simplified equation matches the original

• The equation has no x terms

• The equation has no y terms

In the example y^2 = x^5 - 7x, what happens when you replace y with -y?

• The equation becomes y^2 = x^5 + 7x

• The equation becomes -y^2 = x^5 - 7x

• The equation becomes y^2 = -x^5 + 7x

• The equation remains y^2 = x^5 - 7x

What is the result of simplifying -y^2?

• 2y^2

• 0

• -y^2

• y^2

What is the first step in testing for symmetry around the y-axis?

• Replace all y's with x's

• Replace all x's with negative x's

• Replace all numbers with their negatives

• Replace all variables with zero

In the cubic equation y = x^3 + 4, what happens when you replace x with -x?

• The equation becomes y = -x^3 + 4

• The equation becomes y = -x^3 - 4

• The equation becomes y = x^3 - 4

• The equation remains y = x^3 + 4

What indicates that an equation is not symmetric around the y-axis?

• The equation has no x terms

• The simplified equation has a different sign for x terms

• The equation has no y terms

• The simplified equation matches the original

What is the first step in testing for symmetry around the origin?

• Replace all variables with zero

• Replace all y's with x's

• Replace all x's and y's with their negatives

• Replace all x's with y's

In the rational function y = x / (x^2 + 5), what happens when you replace x and y with their negatives?

• The equation becomes -y = -x / (x^2 + 5)

• The equation becomes -y = x / (x^2 + 5)

• The equation remains y = x / (x^2 + 5)

• The equation becomes y = -x / (x^2 + 5)