Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

ƒ (x) = x^{4} + 5 x^{2} - 12

Accepted Answer

Step 1: Understand the types of symmetry. A function can be symmetric about the x-axis, y-axis, or the origin. For y-axis symmetry, f(x) = f(-x). For x-axis symmetry, if (x, y) is on the graph, then (x, -y) is also on the graph. For origin symmetry, f(-x) = -f(x). Step 2: Check for y-axis symmetry. Substitute -x into the function: f(-x) = $$(-x)^4$$ + $$5(-x)^2 - 12. $$Simplify to see if it equals f(x). Step 3: Simplify f(-x): $$(-x)^4$$ = $$x^4$$ and $$5(-x)^2$$ = $$5x^2$$, so f(-x) = $$x^4$$ + $$5x^2 - 12. $$Since f(-x) = f(x), the function is symmetric about the y-axis. Step 4: Check for origin symmetry. Substitute -x into the function and check if f(-x) = -f(x). We already have f(-x) = $$x^4$$ + $$5x^2 - 12. $$Now, calculate -f(x) = $$-(x^4$$ + $$5x^2 - 12$$) = $$-x^4$$ - $$5x^2 + 12. $$Since f(-x) ≠ -f(x), the function is not symmetric about the origin. Step 5: Conclude that the function f(x) = $$x^4$$ + $$5x^2 - 12 is $$symmetric about the y-axis only. You can verify this by graphing the function and observing its symmetry visually.

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$ƒ (x) = x^{4} + 5 x^{2} - 12$

Verified video answer for a similar problem:

Key Concepts

Symmetry in Graphs

Even and Odd Functions

Graphing Functions

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.ƒ(x)=x4+5x2−12ƒ(x)=x^4+5x^2-12