Skill: 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.