Question

question
move at least one of the 5 guide points below to complete the graph of ( y = (x - 6)^2 + 3 ). 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.
(there is also a coordinate graph with a parabola - like figure, buttons labeled
eset\ and
eflect over x - axis\, and some progress - related text at the top like \deltamath\, \complete: 84%\ etc.)

Explanation:

Step1: Identify the vertex form

The function is \( y=(x - 6)^2+3 \), which is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \((h,k)=(6,3)\). The original graph (blue point is at \((0,0)\) for \( y = x^2 \)) needs to be transformed.

Step2: Move the blue vertex

To get the vertex of \( y=(x - 6)^2+3 \), we shift the blue point (vertex of \( y = x^2 \)) 6 units to the right (because \( h = 6 \)) and 3 units up (because \( k = 3 \)). Then adjust the red points accordingly (since red points are on the parabola, their \( x \)-coordinates will be 6 units right from the original \( y = x^2 \) points, and \( y \)-coordinates will follow the new function's values).

Step3: Check the transformation

For example, a point on \( y = x^2 \) like \((1,1)\) will become \((1 + 6,1+ 3)=(7,4)\) on \( y=(x - 6)^2+3 \), and \((-1,1)\) becomes \((-1 + 6,1 + 3)=(5,4)\), \((2,4)\) becomes \((2+6,4 + 3)=(8,7)\), \((-2,4)\) becomes \((-2 + 6,4+3)=(4,7)\). So we move the blue vertex (originally at \((0,0)\)) to \((6,3)\) and then adjust the red points to these new coordinates.

Answer:

To graph \( y=(x - 6)^2+3 \), move the blue vertex (from \((0,0)\)) 6 units right and 3 units up to \((6,3)\). Then move the red points: for each red point \((x,y)\) on \( y = x^2 \), the new point is \((x + 6,y + 3)\) (e.g., \((1,1)\) becomes \((7,4)\), \((-1,1)\) becomes \((5,4)\), \((2,4)\) becomes \((8,7)\), \((-2,4)\) becomes \((4,7)\)). The final graph has vertex at \((6,3)\) and opens upwards with the red points at the transformed coordinates.