How adding, subtracting, and multiplying change where a graph lives.
A physics teacher drops a ball from a table and projects the path on a screen. The parabola appears centered around the origin. Then she raises the starting height by 3 feet, and the entire curve shifts straight up. The shape of the path does not change — the ball still follows the same arc — but everything moved. That shift is a transformation, and algebra lets you describe it exactly.
Start with a function called f(x). Its graph is whatever shape it is. The transformations below do not change what kind of function it is. They move it, flip it, or stretch it.
The first transformation is a vertical translation. When you add a constant k to the output of a function, the graph shifts up by k units if k is positive, and down by ∣k∣ units if k is negative.
g(x)=f(x)+k
Every point (x,y) on f becomes (x,y+k) on g. The x-coordinate stays put. Only the height changes.
The second transformation is a horizontal translation. When you add k inside the function's input, the graph shifts left by k units if k is positive, and right by ∣k∣ units if k is negative.
g(x)=f(x+k)
This one catches people off guard. Adding inside shifts left. Subtracting inside shifts right. The reason: to get the same output, the input has to compensate in the opposite direction. If the original function needed x=3 to produce a certain value, the new function reaches that same value when x+k=3, meaning x=3−k. The graph moves opposite to the sign of k.
The third transformation is a vertical stretch or compression. Multiplying the output by a constant a pulls the graph away from the x-axis when ∣a∣>1 and squashes it toward the x-axis when 0<∣a∣<1.
g(x)=a⋅f(x)
Every point (x,y) becomes (x,ay). The x-coordinates do not move. The heights scale.
The fourth transformation is a reflection over the x-axis. Multiplying the entire function by −1 flips the graph upside down.
g(x)=−f(x)
Every point (x,y) becomes (x,−y). Points that were above the axis are now below it, and vice versa. Points on the x-axis — where y=0 — do not move.
The graph below shows f(x)=x2 alongside several of its transformations so you can see each effect directly.
Interactive graph — scroll to zoom, drag to pan
The red curve is f(x)=x2. The blue is f(x)+3 — same shape, shifted up 3. The green is f(x−2) — same shape, shifted right 2. The orange is −f(x) — flipped over the x-axis.
Now work through the algebra. Given f(x)=x2, describe each transformation and identify what changed.
Vertical translation: f(x) = x² shifted up 5
g(x)=x2+5
Adding 5 outside the function raises every output by 5.
(0,0)→(0,5)
The vertex was at the origin. Now it is 5 units higher.
Vertex: (0,5)✓
The parabola is identical in shape — just 5 units up.
Horizontal translation: f(x) = x² shifted left 4
g(x)=(x+4)2
Adding 4 inside the input shifts the graph to the left, not the right.
(0,0)→(−4,0)
The vertex moves left by 4 because x + 4 = 0 when x = -4.
Vertex: (−4,0)✓
Positive inside the parentheses means left. It feels backwards — but that is the rule.
Vertical stretch and reflection: g(x) = -3x²
g(x)=−3⋅f(x)
Two things are happening at once: multiplying by 3 stretches, the negative sign flips.
(1,1)→(1,−3)
The point (1,1) on f becomes (1, -3) on g. The x stays. The y gets multiplied by -3.
(2,4)→(2,−12)
Same logic: 4 times -3 is -12. The graph opens downward and is steeper.
Opens down, stretched vertically✓
When the leading coefficient is negative, the parabola opens downward.
Combining transformations: g(x) = 2(x - 1)² + 3
g(x)=2⋅f(x−1)+3
Three transformations stacked: horizontal shift, vertical stretch, vertical shift.
Vertex of f(x)=x2 is (0,0)
Start from the base vertex and apply each transformation.
x−1=0⇒x=1
The horizontal shift moves the vertex to x = 1.
Vertex: (1,3)
The +3 outside lifts the vertex to height 3. The 2 stretches vertically but does not move the vertex.
Opens up, stretched, vertex at (1,3)✓
Positive leading coefficient means it opens upward.
Practice Questions
f(x)=x2. Write the equation for f(x) shifted down 7.
f(x)=x2. Write the equation for f(x) shifted right 3 and up 2.
g(x)=−(x+2)2−1. Describe all transformations from f(x)=x2.
Regents Corner
On Part I and Part II of the Algebra I Regents, transformation problems often give you a graph and ask you to write the equation, or give you an equation and ask you to identify the transformation. For full credit on Part II, you must name both the direction and the magnitude — writing "shifted right 2" earns the point; writing just "horizontal shift" does not.
A student sees g(x) = (x + 5)² and writes "shifted right 5" because the number inside is positive. The shift is actually left 5. Adding inside the input always moves the graph left. To check: ask where the vertex is. x + 5 = 0 when x = -5, so the vertex is at x = -5, which is left of the origin.