Question

question
move at least one of the 5 guide points below to complete the graph of ( y = (x - 3)^2 - 7 ). 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 a coordinate graph with a parabola, guide points, and buttons like reset, reflect over x - axis, show explanation)

Explanation:

Step1: Identify the vertex form

The function is \( y=(x - 3)^2-7 \), which is in vertex form \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h = 3 \), \( k=-7 \), so the vertex should be at \((3,-7)\).

Step2: Analyze the current graph

The current blue vertex is at \((0,0)\). To get the graph of \( y=(x - 3)^2-7 \), we need to move the blue vertex (vertex of the parabola) 3 units to the right (because \( h = 3 \), so shift \( x \)-coordinate from 0 to 3) and 7 units down (because \( k=-7 \), so shift \( y \)-coordinate from 0 to -7). Also, the red points should follow the transformation. For a vertical stretch/compression, if we assume \( a = 1 \) (no stretch/compression), we just need to shift the vertex and the red points accordingly. Moving the blue point (vertex) to \((3,-7)\) and adjusting the red points (which are symmetric around the vertex) to be 1 unit left/right and 1 unit up from the vertex (since for \( y=(x - 3)^2-7 \), when \( x = 2 \) or \( x = 4 \), \( y=(2 - 3)^2-7=1 - 7=-6 \); when \( x = 1 \) or \( x = 5 \), \( y=(1 - 3)^2-7 = 4 - 7=-3 \)) will give the correct graph.

Answer:

To graph \( y=(x - 3)^2-7 \), move the blue vertex (current at \((0,0)\)) to \((3,-7)\). Then move the red points: the ones near \( x = 1 \) and \( x=-1 \) (current) should move to \( x = 4 \) and \( x = 2 \) (with \( y=-6 \)) and \( x = 5 \) and \( x = 1 \) (with \( y=-3 \)) respectively, to reflect the correct parabola. (Note: Since this is a graph - manipulation problem, the key is to shift the vertex to \((3,-7)\) and adjust the red points symmetrically around this new vertex.)