The SAT gives you a function and then asks about g of x equals f of x plus 3, or f of the quantity x minus 2, and wants to know how the graph moved. The rule is short but counterintuitive: changes inside the parentheses move the graph horizontally and in the opposite direction from what the sign suggests.
Changes outside the function move the graph vertically as written, and changes inside move it horizontally in the opposite direction.
Changes outside the function move it vertically in the direction you expect. The graph of f of x plus 3 is the graph of f shifted up 3 units, since every output is 3 larger. The graph of f of x minus 6 is shifted down 6 units. If a question describes a curve moved up or down, look for the constant sitting outside.
Changes inside the parentheses move it horizontally and backward from the sign. The graph of f of the quantity x minus 2 is the graph of f shifted right by 2, not left. Here is why. If the original graph has an interesting point at x equals 0, the new function reaches that same interesting behavior when x minus 2 equals 0, which is at x equals 2. So the feature moved right even though the expression shows a minus.
Combine the two and the description stacks. If f of x equals x squared, then g of x equals the quantity x plus 4 squared minus 1 is the basic parabola shifted 4 units left and 1 unit down, putting its vertex at the point with coordinates negative 4 and negative 1. This is exactly vertex form, which is why the vertex sign rule and the transformation sign rule are the same fact.
A minus sign in front of the whole function reflects it across the x-axis, turning every output into its opposite. So negative f of x flips an upward parabola into a downward one. A minus sign attached to the input, as in f of negative x, reflects across the y-axis instead, swapping left and right.
A number multiplying the whole function stretches it vertically. In 3 times f of x, every output is tripled, so the graph is three times as tall at each input while the zeros stay in place, since tripling zero is still zero. A multiplier between 0 and 1 compresses the graph toward the horizontal axis instead. Testing one convenient input, often x equal to zero, is a fast way to confirm which choice matches.
Reading f of the quantity x minus 2 as a shift to the left because of the minus sign, when the graph actually moves right.
A population doubles every 3 years. If it is 1000 now, what will it be in 12 years?
A patient receives 500 milligrams of a drug, and each hour 8% of the drug still present is cleared. To the nearest milligram, how much remains after 6 hours?
Why: Each period multiplies by 1 - 8/100 = 0.92, so the amount is 500(0.92)^6 ≈ 303. The value 460 applies the decrease a single time, which leaves far too much. Answering 330 means you counted one period too few.
Because the function reaches each of its features at a smaller input. With f of x plus 3, whatever f did at zero now happens when x plus 3 equals zero, which is at x equals negative 3. The feature arrives 3 units earlier, so the whole graph moves left.
It reflects the graph across the x-axis, turning every output into its opposite. Points above the axis go below it by the same amount, and the zeros stay where they were because the opposite of zero is zero.