A transformed function puts a new expression inside a function you already know. For -intercepts, find the original zeros in Desmos, then ask which new makes the inside equal each old zero. A nonzero multiplier outside the function doesn't change its zeros. Watch the inside sign: does not move inputs the same way as .
Hints
- Hint 1
An -intercept is where the output is . Since is nonzero, multiplying by cannot change where the output is . Which inputs make the original function equal ?
- Hint 2
Graph and read every place it crosses the -axis. Those -values are the old zero inputs; the new rule must feed each one into again.
- Hint 3
If is an old zero input, set the whole new input equal to it: . Solve for before adding the new intercepts.
Step-by-step
Map the original zeros
Step 1Find the zeros of the original function
A zero is an input that makes a function's output , so it appears at an -intercept. Type in Desmos and click all three points where the graph meets the -axis. Desmos shows , , and , so the old zero inputs are , , and .
- Step 2
Relate each new intercept to an old zero
At an -intercept of , its output is . Because is nonzero, exactly when ; the leading coefficient doesn't need to be used to find the intercepts. Let be an old zero input. Make the whole input equal the old zero input: .
- Step 3
Solve for the new input
The minus sign before matters. Subtract from both sides: . Multiply both sides by : . So each old zero gives a new intercept at , not .
- Step 4
Map all three zeros
Use for each old zero: gives ; gives ; and gives . The three new -intercepts are at , , and .
- Step 5
Add the new intercepts
The question asks for the sum of all three new -coordinates, not the original zeros. Type in Desmos; it shows . So . Choice D.