Reference Sheet · SAT Math · Advanced Math
Functions & Transformations
Parent functions and the four ways the SAT modifies them: shifts, reflections, and stretches. Each transformation has its own interactive widget so you can see the rule in action.
Parent functions
Six functions show up constantly. Memorize the basic shape of each — every transformation question starts from one of these.
Linear
$$f(x) = x$$
A straight line through the origin with slope $1$. Passes through $(0, 0)$ and $(1, 1)$.
Look closer
Notice: Slope of $1$ — equal rise and run. For every unit you go right, you go up exactly one unit.
Quadratic
$$f(x) = x^2$$
A parabola opening up with vertex at the origin. Passes through $(0, 0)$, $(\pm 1, 1)$, $(\pm 2, 4)$.
Look closer
Notice: Symmetric across the y-axis. Output grows quickly: at $x = 5$, $y$ is already $25$.
Cubic
$$f(x) = x^3$$
An S-shaped curve through the origin. Passes through $(\pm 1, \pm 1)$, $(\pm 2, \pm 8)$.
Look closer
Notice: Point-symmetric through the origin: if $(a, b)$ is on the graph, so is $(-a, -b)$.
Square root
$$f(x) = \sqrt{x}$$
Defined only for $x \geq 0$. Passes through $(0, 0)$, $(1, 1)$, $(4, 2)$, $(9, 3)$.
Look closer
Notice: Defined only for $x \geq 0$ — there's no graph at all to the left of the y-axis.
Absolute value
$$f(x) = |x|$$
A V-shape with vertex at the origin. Always non-negative output.
Look closer
Notice: Two straight lines meeting at a sharp corner at the origin. Output is never negative.
Exponential
$$f(x) = a^x \quad (a > 0)$$
Always positive. Passes through $(0, 1)$ regardless of $a$. Grows when $a > 1$; decays when $0 < a < 1$.
Look closer
Notice: Two cases shown: growth $y = 2^x$ in red, decay $y = (\tfrac{1}{2})^x$ in blue. Both cross the y-axis at $1$.
Vertical shifts
$f(x) + k$ shifts the graph up or down
$$f(x) + k$$
Add $k$ outside the function. If $k > 0$, the graph shifts UP by $k$. If $k < 0$, it shifts DOWN by $|k|$.
EXAMPLE
How does the graph of $g(x) = x^2 + 3$ compare to $f(x) = x^2$?
Same shape, shifted UP by 3.
Vertex moves from $(0, 0)$ to $(0, 3)$.
Try it yourself
- Drag the slider for $k$.
- Watch the red parabola slide up when $k > 0$ and down when $k < 0$.
- The dashed black parabola stays put — that's the original $f(x) = x^2$.
Notice: Adding $k$ outside the function shifts the whole graph vertically by $k$ units. Positive $k$ moves it up.
Horizontal shifts
$f(x - h)$ shifts the graph left or right
$$f(x - h)$$
Subtract $h$ inside the function. If $h > 0$, the graph shifts RIGHT by $h$. If $h < 0$, it shifts LEFT by $|h|$. Counterintuitive: $f(x - 3)$ moves the graph $3$ units RIGHT, not left.
EXAMPLE
How does $g(x) = (x - 5)^2$ compare to $f(x) = x^2$?
Same shape, shifted RIGHT by 5.
Vertex moves from $(0, 0)$ to $(5, 0)$.
Try it yourself
- Drag $h$.
- Watch which way the red parabola slides.
- With $h = 3$ you might expect a left shift, but the parabola moves right. Try it!
Notice: Subtracting $h$ inside the function shifts horizontally by $+h$ — the opposite direction from what the sign suggests. This is the most counterintuitive transformation rule on the SAT.
Reflections
Reflect over x-axis
$$-f(x)$$
Negative on the OUTSIDE. Flips the graph upside down across the x-axis.
EXAMPLE
How does $g(x) = -x^2$ compare to $f(x) = x^2$?
Same shape, flipped upside down.
Opens down instead of up.
Reflect over y-axis
$$f(-x)$$
Negative on the INSIDE. Flips the graph left-to-right across the y-axis.
EXAMPLE
How does $g(x) = \sqrt{-x}$ compare to $f(x) = \sqrt{x}$?
Same shape, flipped to the left side.
Defined for $x \leq 0$ instead of $x \geq 0$.
Look closer
- The black curve is the original $f(x) = \sqrt{x}$.
- The red curve is $-f(x)$ — flipped over the x-axis (negative on the outside).
- The blue curve is $f(-x)$ — flipped over the y-axis (negative on the inside).
Notice: A negative outside the function flips it vertically. A negative inside flips it horizontally. Where the negative sign sits determines which flip you get.
Vertical stretches & shrinks
$cf(x)$ stretches or shrinks vertically
$$cf(x)$$
If $|c| > 1$, the graph stretches vertically (taller and skinnier). If $0 < |c| < 1$, it shrinks (shorter and wider). Negative $c$ also reflects.
EXAMPLE
How does $g(x) = 3x^2$ compare to $f(x) = x^2$?
Same vertex, but 3× as tall at every $x$ value.
At $x = 2$: $f(2) = 4$ but $g(2) = 12$.
Try it yourself
- Drag $c$ above $1$ to make the parabola taller and narrower.
- Drag $c$ between $0$ and $1$ to make it shorter and wider.
- Try negative values for $c$ — what happens to the direction?
Notice: Multiplying by $c$ outside the function stretches vertically by a factor of $c$. Negative $c$ also flips the graph upside down — vertical stretch and reflection rolled into one.