Adding a constant outside the function shifts the graph vertically.
$f(x) = x^2$ (parabola vertex at origin).
$g(x) = x^2 + 3$ shifts the parabola up 3 units (vertex at $(0,3)$).
$h(x) = x^2 - 2$ shifts the parabola down 2 units (vertex at $(0,-2)$).
Adding a constant inside the function (with $x$) shifts the graph horizontally. Warning: The direction is opposite to the sign.
$f(x) = x^2$ (parabola vertex at origin).
$g(x) = (x - 2)^2$ shifts the parabola right 2 units (vertex at $(2,0)$).
$h(x) = (x + 3)^2$ shifts the parabola left 3 units (vertex at $(-3,0)$).
Multiplying the function by a constant $a$ stretches or compresses vertically.
| Condition | Effect | Example |
|---|---|---|
| $a > 1$ | Vertical stretch (taller) | $y = 2x^2$ (parabola narrower) |
| $0 < a < 1$ | Vertical compression (shorter) | $y = \frac{1}{2}x^2$ (parabola wider) |
Multiplying the input by a constant $b$ stretches or compresses horizontally. Warning: The effect is opposite to vertical stretches.
| Condition | Effect | Example |
|---|---|---|
| $b > 1$ | Horizontal compression (squeezed) | $y = \sin(2x)$ (period shorter) |
| $0 < b < 1$ | Horizontal stretch (expanded) | $y = \sin(\frac{x}{2})$ (period longer) |
$f(x) = x^2$ (opens upward).
$g(x) = -x^2$ reflects across the x-axis (opens downward).
$h(x) = \sqrt{x}$ (domain $x \ge 0$).
$j(x) = \sqrt{-x}$ reflects across the y-axis (domain $x \le 0$).
| Transformation | Effect on $y = f(x)$ | Example |
|---|---|---|
| $y = f(x) + k$ | Shift up $k$ units | $x^2 + 2$ |
| $y = f(x) - k$ | Shift down $k$ units | $x^2 - 3$ |
| $y = f(x - h)$ | Shift right $h$ units | $(x-1)^2$ |
| $y = f(x + h)$ | Shift left $h$ units | $(x+1)^2$ |
| $y = a f(x)$, $a > 1$ | Vertical stretch | $2x^2$ |
| $y = a f(x)$, $0 < a < 1$ | Vertical compression | $\frac{1}{2}x^2$ |
| $y = f(bx)$, $b > 1$ | Horizontal compression | $\sin(2x)$ |
| $y = f(bx)$, $0 < b < 1$ | Horizontal stretch | $\sin(x/2)$ |
| $y = -f(x)$ | Reflection across x-axis | $-x^2$ |
| $y = f(-x)$ | Reflection across y-axis | $\sqrt{-x}$ |
When multiple transformations are applied, the order matters. Follow this order:
Describe the transformations of $g(x) = -2(x-3)^2 + 4$ from $f(x) = x^2$.
Final result: Vertex at $(3, 4)$, opens downward, narrower than $x^2$.
Interactive graph below. Click buttons to see different transformations of $f(x) = x^2$:
Describe the transformation from $f(x) = \sqrt{x}$ to $g(x) = \sqrt{x-2} + 3$.
Inside $(x-2)$: shift right 2 units. Outside $+3$: shift up 3 units.
Write the equation for $f(x) = |x|$ shifted left 4 units and down 1 unit.
Left 4: $|x+4|$, down 1: $|x+4| - 1$. So $g(x) = |x+4| - 1$.
Describe the transformations of $h(x) = 3(x+1)^2 - 5$ from $f(x) = x^2$.
Inside $(x+1)$: shift left 1 unit. Outside multiply by 3: vertical stretch by factor 3. Outside $-5$: shift down 5 units.
What transformation changes $f(x) = \sin x$ to $g(x) = \sin(-x)$?
$\sin(-x) = -\sin x$ (since sine is odd). So it's a reflection across the x-axis (or y-axis; both give same result for sine).
Write the equation for $f(x) = \cos x$ compressed horizontally by a factor of 2.
Horizontal compression by factor 2 means multiply input by 2: $g(x) = \cos(2x)$.