Question

graph the inverse of the quadratic $y = (x - 3)^2$ shown in black on the graph.

x	y
2	1
3	0
4	1
5	4

Explanation:

Step1: Recall the method to find the inverse of a relation

To find the inverse of a relation (given by a table of values), we swap the \( x \)-values and the \( y \)-values. That is, if a point \((a,b)\) is on the original graph, then the point \((b,a)\) will be on the graph of the inverse relation.

Step2: Swap \( x \) and \( y \) values for each point

• For the point \((1,4)\) (original), the inverse point is \((4,1)\).
• For the point \((2,1)\) (original), the inverse point is \((1,2)\).
• For the point \((3,0)\) (original), the inverse point is \((0,3)\).
• For the point \((4,1)\) (original), the inverse point is \((1,4)\) (wait, no, wait: original point \((4,1)\), so inverse is \((1,4)\)? Wait, no, let's correct. Wait, original table:
• When \( x = 1 \), \( y = 4 \)
• When \( x = 2 \), \( y = 1 \)
• When \( x = 3 \), \( y = 0 \)
• When \( x = 4 \), \( y = 1 \)
• When \( x = 5 \), \( y = 4 \)

So to find the inverse, we create a new table where we take \( y \) as the new \( x \) and \( x \) as the new \( y \).

So:

• Original \((x,y)=(1,4)\) becomes \((y,x)=(4,1)\)
• Original \((2,1)\) becomes \((1,2)\)
• Original \((3,0)\) becomes \((0,3)\)
• Original \((4,1)\) becomes \((1,4)\)
• Original \((5,4)\) becomes \((4,5)\)

Now, we can plot these points \((4,1)\), \((1,2)\), \((0,3)\), \((1,4)\), \((4,5)\) to graph the inverse.

Answer:

To graph the inverse, swap the \( x \)- and \( y \)-values of each point from the original table (e.g., \((1,4)\) becomes \((4,1)\), \((2,1)\) becomes \((1,2)\), etc.) and plot the new points. The inverse relation's graph will be the reflection of the original quadratic's graph over the line \( y = x \).