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.
$\text{ƒ}\left(x\right)=x^4+5x^2-12$

Verified step by step guidance

1. 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).
2. 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).
3. 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.
4. 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.
5. 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.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the y-axis if replacing x with -x yields the same function value, indicating even symmetry. It is symmetric about the x-axis if replacing y with -y gives the same x-value, and it is symmetric about the origin if replacing both x and y with their negatives results in the same function.

Even and Odd Functions

Even functions are defined by the property f(-x) = f(x), which indicates symmetry about the y-axis. Odd functions satisfy f(-x) = -f(x), showing symmetry about the origin. Understanding whether a function is even, odd, or neither helps in determining the type of symmetry present in its graph, which is crucial for analyzing the given function.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x) and output (f(x)). This visual representation aids in identifying properties such as symmetry, intercepts, and overall shape. Using graphing tools or software can enhance accuracy and provide a clearer understanding of the function's behavior, especially when checking for symmetry.