Worked examples
The questions the video works, written out: the setup, each step, the answer and the trap.
0:44Example 1
The graph of is a V with its corner at (0, 0). Where is the corner, and which way does the graph open, for each of , , and ?
- . The +2 sits outside the function, so it moves the graph vertically, exactly as written: up 2. Corner (0, 2), still opening up. Check: at x = 0, |0| + 2 = 2.
- . The -3 sits inside, attached to x, so it moves the graph horizontally and does the opposite of what it looks like: right 3. Corner (3, 0). Check: at x = 3, |3 - 3| = 0, and at x = 4, |1| = 1.
- . The negative is outside, so it flips the graph over the x-axis. The corner stays at (0, 0) but the V opens downward. Check: at x = 2, -|2| = -2.
Answer: f(x) + 2: corner (0, 2), opens up. f(x - 3): corner (3, 0), opens up. -f(x): corner (0, 0), opens down.
Outside the function means vertical and as written; inside means horizontal and opposite. The trap that leaks the most points is f(x - 3), which students slide left to (-3, 0); the corner lands where the inside equals zero, x - 3 = 0, so x = 3. A second trap is confusing -f(x), a flip over the x-axis, with f(-x), a flip over the y-axis.
1:15Example 2
Why does the graph of move right, not left? Find the x-coordinate of its corner without plotting any points.
- The corner of f(x) = |x| happens where the input to the absolute value is 0. For f(x - 3) the input is x - 3, so the corner happens where x - 3 = 0.
- Solve: x = 3. The corner has moved to x = 3, three units to the right of the original corner at x = 0.
- Check with a matching output: the original has f(1) = 1. The new graph produces that same output when x - 3 = 1, that is at x = 4, again three units to the right. Every point rides 3 to the right.
Answer: the corner is at x = 3; the graph moves right 3
Subtracting inside means each output now needs an x that is 3 larger to produce the same input, so the whole picture shifts right. The trap is treating the inside like the outside and reading a minus sign as left; the inside always reverses the sign you see.
1:45Example 3
Using the same f(x) = |x|, where is the corner of ?
- Split the inside from the outside. Inside: x + 2. Set it to zero to find the x-coordinate of the corner: x + 2 = 0 gives x = -2. A plus inside means left 2.
- Outside: -3. It is outside, so it moves the graph vertically as written: down 3. The y-coordinate of the corner is 0 - 3 = -3.
- Corner (-2, -3), and the graph still opens upward because there is no negative in front of the absolute value.
- Check: . Two units either side: and , the same height, so (-2, -3) is the symmetric corner.
Answer: corner (-2, -3)
The inside x + 2 moves the graph left 2, the opposite of the sign, and the outside -3 moves it down 3, as written. The wrong answer (2, -3) comes from sliding right because the sign was plus, and (-2, 3) comes from flipping the outside sign too; only the inside reverses.
Lesson transcript
The narration of the video, word for word, under its chapter headings.
Welcome to SAT Climb. The SAT loves to take one graph and shift it, slide it, or flip it — then ask you to find the new picture. You never re-plot a single point. You just move what you already have. Start with a base function — call it f. Here f(x) = |x|: a clean V with its corner at the origin. Every transformation question wraps f in something new. Watch what each piece does — the same rules work for any function. Three moves cover almost everything. Add outside — f(x) + 2 — and the V lifts up 2. Subtract inside, next to x — f(x − 3) — and it slides right 3. A negative in front — −f(x) — flips it over the x-axis. Up, over, flip. Here's the rule that ends the guessing. Outside the function moves it vertically, and it behaves — +2 means up 2. Inside, attached to x, moves it horizontally, and does the opposite. −3 moves right, not left. Why? Which x makes the inside zero? For x − 3, that's x = 3 — the corner rides to 3. Your turn. Same f = |x|. Where does the corner land for g(x) = |x + 2| − 3? Split it. Inside x + 2 moves it left 2. Outside − 3 drops it down 3. New corner: (−2, −3). Three traps. One: f(x − h) moves right, not left. Two: + k is outside, so it's vertical — never sideways. Three: −f(x) flips over the x-axis, but f(−x) flips over the y-axis. Outside vs inside, every time. Transformations: solved. Read outside for vertical, inside for the opposite horizontal, and you'll place any graph without plotting a point. Start your free trial at satclimb.com. Keep climbing.
Read the written version: the Nonlinear functions strategy guide, then try 25 hard Nonlinear functions questions with full explanations. Both are free, no account needed.