Identify reflections across the x-axis

A reflection across the x-axis maps each point $(x,y)$ to $(x,-y)$ : the x-coordinate is preserved while the y-coordinate, or function output, changes sign. Thus the graph of $y=f(x)$ becomes $y=-f(x)$ ; positive and negative outputs switch, while x-intercepts remain fixed. This distinguishes x-axis reflection from y-axis reflection, which changes inputs according to $f(-x)$ .

Detailed Explanation: Identify reflections across the x-axis

To reflect a graph across the $x$ -axis, keep each $x$ -coordinate the same and change the sign of each $y$ -coordinate:

(x,y)\longrightarrow (x,-y)

For a function $y=f(x)$ , replace the function with:

y=-f(x)

Example

Reflect the function

f(x)=x^2-4x+3

across the $x$ -axis.

Step 1: Change the sign of the entire function.

y=-f(x)

Step 2: Substitute the expression for $f(x)$ .

y=-(x^2-4x+3)

Step 3: Distribute the negative sign.

\boxed{y=-x^2+4x-3}

So, the reflected function is:

\boxed{y=-x^2+4x-3}

For example, the original graph contains the point $(2,-1)$ because $f(2)=-1$ . After reflection, it becomes $(2,1)$ . The $x$ -coordinate stays $2$ , while the $y$ -coordinate changes sign.

Any point on the $x$ -axis, such as an $x$ -intercept, stays in the same place because its $y$ -coordinate is $0$ . Be careful not to use $f(-x)$ ; that changes the input and represents a reflection across the $y$ -axis.

Learn by doing: Identify reflections across the x-axis

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Function Transformations (Definition) - Double Definition (Values) to Transformation

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