A sequence of graph transformations changes a function rule one move at a time. Reflecting across the -axis changes the input to its opposite; shifting right then changes the input of that reflected rule. Keep the moves in the given order, and use Desmos to evaluate the original function once you've found the input it needs. A minus sign outside the function would reflect heights instead of horizontal positions.
Hints
- Hint 1
A reflection across the -axis reverses each point's horizontal position, so the new rule uses the opposite input, . A minus sign outside would reflect across the -axis instead.
- Hint 2
For a translation units right, replace with in the rule you have after reflecting. Keep in parentheses when you put it inside .
- Hint 3
A shift down changes the output, so subtract outside . To find , first work out which input goes into ; then evaluate that output and subtract .
Step-by-step
Build the transformed rule, then evaluate
Step 1Reflect the original graph
A reflection across the -axis sends each -coordinate to its opposite, so the first rule is . The minus sign goes inside ; would reflect across the -axis.
- Step 2
Shift the reflected graph right
A right shift of replaces with in the rule you already have: . The shift acts on the reflected rule, so the minus sign applies to the whole .
- Step 3
Shift the output down
Moving down subtracts from each output, so . Keep the outside : it changes the height, not the input.
- Step 4
Find the input that reaches the original function
Put in for to find : . Work out the input inside : . So you need the original function's output at , followed by the downward shift.
- Step 5
Evaluate the original function
To evaluate , put into every factor. Type in Desmos, then type . Desmos shows , the output before the graph moves down.
- Step 6
Apply the final shift
Type on the next Desmos line. It shows , so the new graph's height at is . Grid in -43.