Apply reflections across coordinate axes

Reflection across the $x$ -axis maps $(x,y)$ to $(x,-y)$ and transforms $y=f(x)$ into $y=-f(x)$ , while reflection across the $y$ -axis maps $(x,y)$ to $(-x,y)$ and transforms it into $y=f(-x)$ . The distinction between changing input signs and output signs supports accurate interpretation and composition of function transformations; this scope does not include reflections across arbitrary lines or in higher dimensions.

Detailed Explanation: Apply reflections across coordinate axes

To reflect a function’s graph, use the axis of reflection to decide which sign changes:

Across the $x$ -axis: change the output sign.

(x,y)\mapsto(x,-y),\qquad y=f(x)\mapsto y=-f(x)

Across the $y$ -axis: change the input sign.

(x,y)\mapsto(-x,y),\qquad y=f(x)\mapsto y=f(-x)

Worked example

Let

f(x)=x^2-3x+2.

Find the equations of the graphs reflected across the $x$ -axis and the $y$ -axis.

1. Reflection across the $x$ -axis

Change the sign of the entire function:

y=-f(x)

Substitute the function:

y=-(x^2-3x+2)

Distribute the negative sign:

\boxed{y=-x^2+3x-2}

2. Reflection across the $y$ -axis

Replace every $x$ with $(-x)$ :

y=f(-x)

Substitute $(-x)$ into the function:

y=(-x)^2-3(-x)+2

Simplify:

\boxed{y=x^2+3x+2}

Remember: reflection across the $x$ -axis changes output signs, while reflection across the $y$ -axis changes input signs.

Learn by doing: Apply reflections across coordinate axes

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Cartesian Grid - Reflection of Point (Coordinates to Coordinates) across Axis

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