Unit 1, Day 2: Translations (Shifts)
October 28, 2025
Today's Goal
I will be able to move (or "translate") a parent function up, down, left, or right by changing its equation.
Warm-Up: Quick Review
What are the three parent functions we learned yesterday?
Match the name to the equation:
1. Linear
2. Quadratic
3. Exponential
A) $f(x) = x^2$
B) $f(x) = 2^x$
C) $f(x) = x$
I Do: Vertical Shifts (Up & Down)
To move a graph up or down, we add or subtract a number at the very end of the equation, outside of any parentheses.
$f(x) + k$
- $f(x) = x^2 + 3$: Moves the graph UP 3 units.
- $f(x) = x^2 - 5$: Moves the graph DOWN 5 units.
- See it: In Desmos, type $y=x^2$ and then $y=x^2+3$.
We Do: Horizontal Shifts (Left & Right)
To move a graph left or right, we add or subtract a number inside parentheses, right next to the $x$.
$f(x-h)$
Watch out! This one is tricky. It moves the OPPOSITE direction of the sign!
- $f(x) = (x-4)^2$: Moves the graph RIGHT 4 units.
- $f(x) = (x+2)^2$: Moves the graph LEFT 2 units.
- Let's see it together: In Desmos, type $y=x^2$ and then $y=(x-4)^2$.
You Do: Combining Shifts
What happens when we put it all together?
$f(x) = (x+1)^2 - 5$
Your turn to describe the two shifts:
- The $(x+1)$ inside the parentheses moves the graph... which way?
- The $-5$ at the end of the equation moves the graph... which way?
- Check your work: Graph the parent $y=x^2$ and then graph $y=(x+1)^2-5$ in Desmos. Did it move where you expected?
Independent Practice
Describe the shift for each equation.
Green Level
1. $f(x) = x+10$
2. $f(x) = x^2 - 1$
3. $f(x) = (x+5)$
Yellow Level
4. $f(x) = (x-2)^2+3$
5. $f(x) = 2^x - 8$
6. $f(x) = (x+6)^2 - 4$
Red Level
7. Write an equation for a quadratic function moved left 1 and down 9.
8. Write an equation for a linear function moved up 20.
Exit Ticket
On a separate piece of paper, answer the following.
How do you change the equation of $f(x)=x^2$ to move it down 7 units?
Describe the shift in the graph of $f(x)=(x+10)^2$. (Which way does it move?)
Write the new equation for the parent $f(x)=2^x$ after it has been moved right 3 and up 6.