Reflecting Functions Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Given

f(x)=x^3-2

, write the equations for (a) reflection over the

x

-axis and (b) reflection over the

y

-axis. Evaluate each at

x=2

1
(a) Reflection over $x$ -axis: negate the output → $g(x)=-f(x)=-(x^3-2)=-x^3+2$ . $g(2)=-8+2=-6$ .
2
(b) Reflection over $y$ -axis: negate the input → $h(x)=f(-x)=(-x)^3-2=-x^3-2$ . $h(2)=-8-2=-10$ .
3
Note: $f(2)=8-2=6$ ; the $x$ -axis reflection negates the output ( $-6$ ); the $y$ -axis reflection changes the sign of $x$ ( $-10$ ).

Answer

(a)

g(x)=-x^3+2

g(2)=-6

; (b)

h(x)=-x^3-2

h(2)=-10

Two fundamental reflections:

-f(x)

flips the graph over the

x

-axis (negates all outputs);

f(-x)

flips over the

y

-axis (reverses all inputs). They are distinct transformations that generally produce different results.

About Reflecting Functions

Reflecting a function mirrors its graph across the $x$ -axis ( $-f(x)$ ), $y$ -axis ( $f(-x)$ ), or the line $y = x$ (the inverse function).

Show that [formula] is unchanged by reflection over the [formula]-axis (even function) but [formula]

The point [formula] is on the graph of [formula]. Give the corresponding point on: (a) [formula], (b

Classify [formula] and [formula] as even, odd, or neither. Explain using the definitions.