Tip #44 to Get a Top SAT Math Score

Transformations (and functions) are the only topics from Algebra II on the SAT. Transformations begin with a "parent" equation or graph, for example, y = x². It's called the parent because the transformed graphs are birthed/derived from it. A transformation happens when we add, subtract, multiply, or divide a number into the original that causes it to move it up, down, left, or right. It might even make it skinnier or fatter.

Here are all the transformations that the SAT uses:

y = x² + 1 moves every point of the graph up 1 unit.

y = x² – 1 moves every point of the graph down 1 unit.

y = (x +1)² moves every point of the graph left 1 unit.

y = (x – 1)² moves every point of the graph right 1 unit.

y = 2(x)² makes the graph skinnier.

y = 0.5(x)² makes the graph fatter.

y = –(x)² reflects the graph across the x axis.

Let's look at this question:

Solution: This is a straightforward transformation. The parent function g(x) needs to be shifted 1 unit to the left. If you memorized the transformations shown above, this question is easy; a left transformation would be y = g(x + 1). If you did not memorize the transformations, well … get to it, you silly slacker. Of course, you could also graph each choice on your calculator and see which one yields results that are all 1 unit to the left of the parent.

Correct answer: B

Example Problems

Easy

1. Which of the following could be the equation f the graph above?
   1. y = x – 1
   2. y = x² –1
   3. y = (x – 1)²
   4. y = (x +1)²
   5. y = x²

Medium

2. The graph of g(x) is shown above. Which of the following equations represents all the points of g(x) shifted 1 unit to the right?
   1. y = g(x) +1
   2. y = g(x + 1)
   3. y = g(x – 1)
   4. y = g(x) + 1
   5. y = g(x)

Hard

3. Which of the following could represent all the points of the above graph moved 2 units down?
   1. y = x – 2
   2. y = x² – 2
   3. y = (x – 1)² – 2
   4. y = (x +1)² – 2
   5. y = (x – 2)²



4. The figure above shows the graphs of the functions m and n. The function m is defined by m(x) = x³ + 2x, and the function n is defined by n(x) = m(x + h) – k, where h and k are constants. What is the value of h – k ?
   1. 0
   2. 1
   3. 2
   4. 3
   5. 4