Unit 1, Day 3: Reflections (Flips)
October 29, 2025
Today's Goal
I will be able to flip (or "reflect") a parent function over the x-axis or y-axis by changing its equation.
Warm-Up: Describe the Shift
How does the graph of the equation below move from its parent function, $f(x)=x^2$?
$f(x) = (x-5)^2 + 1$
Hint: There are two moves. Think about what the number inside the parentheses does, and what the number at the end does.
I Do: Reflecting over the x-axis
To flip a graph upside down (over the x-axis), we put a negative sign in the very front of the equation.
$-f(x)$
- The equation $f(x) = -x^2$ flips the parabola upside down.
- See it: In Desmos, type $y=x^2$ and then $y=-x^2$.
We Do: Reflecting over the y-axis
To flip a graph sideways (over the y-axis), we put a negative sign inside the parentheses, attached to the $x$.
$f(-x)$
- The equation $f(x) = (-x+2)^2$ flips the parabola sideways.
- Let's see it together: In Desmos, type $y=(x-2)^2$ to see the original. Then, in a new line, type $y=(-x-2)^2$ to see the flip.
You Do: Combining Flips & Shifts
Let's combine what we've learned.
$f(x) = -(x+3)^2$
Your turn to describe the two transformations:
- What does the negative sign in the front do?
- What does the $(x+3)$ inside the parentheses do?
- Check your work: Graph the parent $y=x^2$ and then graph $y=-(x+3)^2$ in Desmos. Did it move and flip where you expected?
Independent Practice
Describe the reflection for each equation.
Green Level
1. $f(x) = -x$
2. $f(x) = -x^2$
3. $f(x) = -(x+5)^2$
Yellow Level
4. $f(x) = -2^x$
5. $f(x) = (-x-1)^2$
6. Describe two transformations for $f(x) = -x^2+3$.
Red Level
7. Write an equation for a quadratic function flipped over the x-axis and moved right 2.
8. What happens to the graph of $f(x)=-(-x)^2$?
Exit Ticket
On a separate piece of paper, answer the following.
Does a negative sign in the front of an equation flip the graph over the x-axis or y-axis?
Write the new equation for the parent $f(x)=x^2$ after it's been reflected over the x-axis and moved up 4 units.