Function Transforms

Function Transformations

This pattern tests whether you understand how replacing $x$ with $(x - h)$ or $(x + h)$ shifts a function's graph horizontally. The SAT focuses heavily on finding the new vertex location or new $x$-intercepts after a shift.

The Key Rule

$g(x) = f(x - h)$ shifts the graph of $f$ to the right by $h$ units.
$g(x) = f(x + h)$ shifts the graph of $f$ to the left by $h$ units.

This is counterintuitive: the sign inside the function is opposite to the direction of the shift.

Every feature of the graph shifts by the same amount: vertex, $x$-intercepts, and all other points.

Worked Examples

Example 1. $f(x) = 2x^2 + 24x + 10$ and $g(x) = f(x - 3)$. At what value of $x$ does $g$ reach its minimum?

A) $-9$
B) $-6$
C) $-3$
D) $3$

Step 1: Find the vertex of $f$. The $x$-coordinate of the vertex is $x = -\dfrac{b}{2a} = -\dfrac{24}{2(2)} = -6$.
Step 2: Since $g(x) = f(x - 3)$, the graph shifts right by $3$.
New vertex: $-6 + 3 = -3$.
Gotcha: Option B ($-6$) is the vertex of $f$, not $g$. Option A ($-9$) shifts left instead of right. Remember: $f(x - 3)$ shifts right.
The answer is C.

Example 2. $f(x) = 3x^2 - 36x + 50$ and $g(x) = f(x + 2)$. At what value of $x$ does $g$ reach its minimum?

A) $4$
B) $6$
C) $8$
D) $-2$

Vertex of $f$: $x = -\dfrac{-36}{2(3)} = \dfrac{36}{6} = 6$.
$g(x) = f(x + 2)$ shifts left by $2$: $6 - 2 = 4$.
The answer is A.

Example 3. $f(x) = (x - 2)(x + 3)(x + 4)$ and $g(x) = f(x + 2)$. What is the sum of the $x$-intercepts of $g$?

A) $-11$
B) $-5$
C) $-1$
D) $1$

The $x$-intercepts of $f$ are at $x = 2, -3, -4$ (where each factor is zero).
Sum of $f$'s intercepts: $2 + (-3) + (-4) = -5$.
$g(x) = f(x + 2)$ shifts left by $2$. Each intercept shifts left by $2$:
$2 - 2 = 0$, $-3 - 2 = -5$, $-4 - 2 = -6$.
Sum: $0 + (-5) + (-6) = -11$.
Shortcut: If each of $n$ intercepts shifts by $-2$, the sum decreases by $2n$. Original sum $= -5$, three intercepts shift by $-2$ each: $-5 + 3(-2) = -5 - 6 = -11$.
The answer is A.

Example 4. $h(x) = (x+1)(x+2)(x-5)$ and $k(x) = h(x - 3)$. What is the sum of the $x$-intercepts of $k$?

Intercepts of $h$: $x = -1, -2, 5$. Sum $= 2$.
$k(x) = h(x - 3)$ shifts right by $3$: each intercept $+ 3$.
New intercepts: $2, 1, 8$. Sum $= 11$.
Shortcut: $2 + 3(3) = 2 + 9 = 11$.

Example 5. The graph of $y = f(x)$ is a parabola with vertex at $(4, -2)$. The graph of $y = f(x) + 5$ shifts the graph up by $5$. What is the new vertex?

Adding $5$ outside the function shifts the graph up by $5$. The $x$-coordinate doesn't change; the $y$-coordinate increases by $5$.
New vertex: $(4, -2 + 5) = (4, 3)$.

What to Do on Test Day

• $f(x - h)$ = shift RIGHT by $h$. $f(x + h)$ = shift LEFT by $h$. The sign is opposite to the direction. This is the most-tested rule.
• $f(x) + k$ = shift UP by $k$. $f(x) - k$ = shift DOWN by $k$. Vertical shifts follow the sign.
• Find the original vertex first using $x = -\dfrac{b}{2a}$, then apply the shift.
• All features shift together. If the vertex moves right by $3$, every $x$-intercept also moves right by $3$.
• Sum of intercepts shortcut: If $n$ roots each shift by $d$, the new sum $=$ old sum $+ n \cdot d$.
• Don't second-guess the direction. The minus sign in $f(x - 3)$ feels like "left" but it's right. Trust the rule.