Graph Symmetry and Equations

The video tutorial explains how to determine the symmetry of graphs about the x-axis, y-axis, and origin using algebraic and visual methods. It covers three main tests for symmetry and provides examples with polynomial, ellipse, and complex graphs. The tutorial emphasizes the importance of verifying symmetry both visually and algebraically, demonstrating each method step-by-step.

What is the first step to test for x-axis symmetry in a graph?

• Replace x with -x

• Replace y with -y

• Replace both x and y with their negatives

• Replace y with x

In the graph of y = x^4 + 4x^2 + 5, which type of symmetry is present?

• x-axis symmetry

• y-axis symmetry

• Origin symmetry

• No symmetry

Why does the graph of y = x^4 + 4x^2 + 5 not have x-axis symmetry?

• The equation is equivalent when both x and y are replaced with their negatives

• The equation is not equivalent when x is replaced with -x

• The equation is not equivalent when y is replaced with -y

• The graph is not symmetrical visually

What type of symmetry does the graph of y = x^3 + 2x exhibit?

• No symmetry

• Origin symmetry

• y-axis symmetry

• x-axis symmetry

How can you verify origin symmetry algebraically for y = x^3 + 2x?

• Replace x with -x only

• Replace both x and y with their negatives and check for equivalence

• Replace y with -y only

• Replace x with -x and y with -y, then multiply the equation by -1

What shape does the equation x^2 + 4y^2 = 4 represent?

• A hyperbola

• A parabola

• A circle

• An ellipse

Which symmetries are present in the graph of x^2 + 4y^2 = 4?

• x-axis and y-axis symmetry only

• x-axis, y-axis, and origin symmetry

• y-axis and origin symmetry only

• No symmetry

For the graph of y^2 - 2y = 4x, which symmetry is verified visually?

• y-axis symmetry

• Origin symmetry

• x-axis symmetry

• No symmetry

Why does the graph of y^2 - 2y = 4x not have y-axis symmetry?

• The graph is symmetrical visually

• The equation is equivalent when x is replaced with -x

• The equation is not equivalent when x is replaced with -x

• The equation is equivalent when y is replaced with -y

What is the result of replacing both x and y with their negatives in the equation y^2 - 2y = 4x?

• The equation becomes a different type of symmetry

• The equation remains unchanged

• The equation becomes non-equivalent to the original

• The equation becomes equivalent to the original