Every move changes the rule in one place. A change outside the brackets acts on heights. A change inside acts on x, and acts backwards. Drag the sliders and watch the gray parent shape turn into the blue one.
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Add a number at the end of the rule and every height goes up by that number: y = x² + 2 is the parabola x² lifted by 2. A negative number lowers it.
Nothing else changes. The shape, the width and the left-right position stay exactly as they were. This is a vertical shift.
Changes outside the brackets act on heights: add 2 and every point goes up 2.
Change the inside and the shape slides sideways, and it goes backwards: y = (x − 3)² moves 3 to the right, not left.
The bottom of the bowl is where the inside equals zero. For (x − 3) that happens at x = 3. Moving the wrong way is the most common slip with the sign of h.
Changes inside act on x, and act backwards: (x − 3) slides the shape 3 to the right.
A minus in front flips the shape upside down: y = −√x. A minus on x flips it left to right: y = √(−x).
On x² you cannot see the second flip, because x² is already symmetric. The square root has only one side, so it shows both flips clearly.
A minus in front flips the heights. A minus on x flips left and right.
A number in front pulls the shape taller (bigger than 1) or flatter (between 0 and 1): y = 2x² looks narrower, not wider.
A number on x acts sideways and backwards: y = (2x)² squeezes the shape toward the y-axis. Telling the two apart, and doing several moves in the right order, is the skill.
A number in front pulls the shape taller or flatter. A number on x acts sideways, and backwards.
The bowl reaches its bottom when the inside equals zero. For (x − 3)² that happens at x = 3, so the whole graph shifts 3 to the right. Changes inside the brackets always act backwards.
A vertical shift adds a number outside the function, like f(x) + 2, and moves the graph up. A horizontal shift changes the input, like f(x − 2), and moves it right. Outside changes act on heights, inside changes act on x.
Put a minus in front, −f(x), to flip it over the x-axis. Put a minus on x, f(−x), to flip it over the y-axis.
In the same order you would calculate the rule: the inside first, then the multiplier, then the number added at the end. For y = −2(x − 1)² + 3: shift right 1, stretch by 2 and flip, then shift up 3.
World 4 of Pip: Math Adventure is about transformations. You slide, flip and stretch a shape so Pip lands on the gems, and you read the moves from the equation.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.