Question

move at least one of the 5 guide points below to complete the graph of $y=(x - 1)^2$. moving the red points changes the vertical stretch or compression. moving the blue point shifts the function left/right/up/down. click the buttons below to start over or reflect over the x - axis.
reset reflect over x - axis

Explanation:

Step1: Identify the vertex of \( y=(x - 1)^2 \)

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. For \( y=(x - 1)^2 \), \( h = 1 \), \( k = 0 \), so the vertex is \((1,0)\). Currently, the blue vertex is at \((0,0)\), so we need to move the blue point to \((1,0)\).

Step2: Check other points (optional, but for accuracy)

For \( y=(x - 1)^2 \), when \( x = 0 \), \( y=(0 - 1)^2 = 1 \); when \( x = 2 \), \( y=(2 - 1)^2 = 1 \); when \( x = -1 \), \( y=(-1 - 1)^2 = 4 \); when \( x = 3 \), \( y=(3 - 1)^2 = 4 \). So the red points should be adjusted to match these coordinates. But the key first step is moving the blue vertex to \((1,0)\) to shift the parabola right by 1 unit (since the original \( y = x^2 \) has vertex at \((0,0)\), and \( y=(x - 1)^2 \) is a horizontal shift right by 1).

Answer:

Move the blue vertex (the lowest point of the parabola) from \((0,0)\) to \((1,0)\). Then adjust the red points: for example, the red points at \( x = -1 \) and \( x = 3 \) should have \( y = 4 \), and the red points at \( x = 0 \) and \( x = 2 \) should have \( y = 1 \) to match \( y=(x - 1)^2 \). (The main action to start is moving the blue point to \((1,0)\).)