Which statement describes a property of an even function?

Which statement describes a property of an even function?

• A. Symmetric across the y-axis
• B. Only even exponents
• C. Can shift up or down, but not left or right
• D. The graph passes through the origin

Answer: (A)

Even functions have symmetry about the y-axis: for every x, the value at x equals the value at -x, so the graph mirrors left to right. This mirror property, f(-x) = f(x), is what makes a function even, and it’s why the statement about symmetry across the y-axis is the best description. For instance, f(x) = x^2 satisfies f(-x) = (-x)^2 = x^2. Note that having only even exponents is a common way to build even functions, but it isn’t required—functions like |x| or cos x are also even without relying on a polynomial with even powers. Shifting vertically keeps the symmetry about the y-axis, but shifting left or right generally destroys that symmetry unless the shift is zero. And whether the graph passes through the origin isn’t a defining feature of evenness; some even functions go through the origin, others do not.

Even functions have symmetry about the y-axis: for every x, the value at x equals the value at -x, so the graph mirrors left to right. This mirror property, f(-x) = f(x), is what makes a function even, and it’s why the statement about symmetry across the y-axis is the best description. For instance, f(x) = x^2 satisfies f(-x) = (-x)^2 = x^2.

Note that having only even exponents is a common way to build even functions, but it isn’t required—functions like |x| or cos x are also even without relying on a polynomial with even powers. Shifting vertically keeps the symmetry about the y-axis, but shifting left or right generally destroys that symmetry unless the shift is zero. And whether the graph passes through the origin isn’t a defining feature of evenness; some even functions go through the origin, others do not.