Function Transformations on the SAT
Shifts, Stretches, and Reflections
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Function Transformations
Function transformations are the SAT way of asking, how does an equation change the graph. If you can read shifts and stretches quickly, you can match equations to graphs without drawing a full picture.
This lesson shows how changes inside parentheses move a graph left or right, and how changes outside move it up or down or stretch it. You will also see why the signs can feel reversed and how to keep them straight.
A Simple Definition Unlocks Function Transformations
Start with a parent function like $f(x) = x^2$ or $f(x) = |x|$. Then compare $g(x) = (x - 2)^2$ and $h(x) = (x + 2)^2$. The graph moves right by $2$ for $(x - 2)$ and left by $2$ for $(x + 2)$.
Multiplying outside the function changes the vertical stretch and can reflect the graph. A negative sign in front reflects across the $x$ axis, and a factor greater than $1$ makes the graph steeper.
Work Through Function Transformations Step by Step
Starting with $f(x) = x^2$, write the function after shifting right $2$, up $1$, then reflecting across the $x$-axis.
Identify the transformation from the equation by comparing it to the parent function.
Start with the base function
Shift right $2$ and up $1$
Reflect across the $x$ axis
Use Desmos to Check Function Transformations
Starting with $f(x) = x^2$, write the function after shifting right $2$, up $1$, then reflecting across the $x$-axis.
Desmos makes transformations visual. Graph the base function and the transformed function at the same time to compare movement.
f(x) = x^2
g(x) = (x - 2)^2 + 1
Algebra is faster for identifying the rule, but Desmos is great for confirming direction and distance.
Desmos is faster for visualizing shifts and reflections. Algebra is faster when you are solving for parameters from an equation.
Expert move: Graph the key functions together and use intercepts or a table to compare how values change; Desmos makes the shape (linear vs exponential, shifts, stretches) obvious.
Reminder: Desmos confirms the picture, but you still have to interpret axes, units, and context.
- Desmos features used: graphing, multiple expressions.
- Common mistake: reversing the direction of a horizontal shift.
Practice Function Transformations with SAT-Style Questions
Match each transformation to its effect.
If shifts right , which expression represents the new function?
What transformation does represent?
Which transformation turns into ?
If , which equation represents a vertical stretch by factor ?
Key Takeaways to Remember for Function Transformations
- Inside changes move left or right, outside changes move up or down.
- Negative outside reflects across the $x$ axis, negative inside reflects across the $y$ axis.
- Desmos helps visualize transformations quickly.
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