The function is defined by
The graph of is shifted units to the left and units down to obtain the graph of . Which of the following equations represents ?
A translation moves a graph without changing its shape. When a quadratic is in vertex form, its turning point is visible, so you can use it to check the shift on Desmos. A left shift adds to the input, while a downward shift subtracts from the output. The trap is making the inside sign match the word “left.”
Hints
- Hint 1
In vertex form , the turning point is because the square is zero when . Find that point in the given function so you can check where the shifted graph should land.
- Hint 2
A left shift changes the input: use inside . A downward shift changes the output, so subtract after evaluating the function. How can you write both changes in one rule?
Step-by-step
Shift the function, then simplify
Step 1Locate the original turning point
Type in Desmos and click its vertex, the parabola’s turning point. Desmos shows . That fits the rule: the square is zero when , leaving an output of .
- Step 2
Translate both shifts into one rule
Type on the next line. To shift left, add inside the input; to shift down, subtract outside the function. The inside sign can feel backward: at the new input , , the old vertex’s input. Click ’s vertex; Desmos shows .
- Step 3
Replace the input in the given rule
Put the whole new input wherever appears in , then keep the downward shift outside:
- Step 4
Simplify inside the square
Combine the numbers inside the square:
- Step 5
Finish the shifted equation
Subtract from outside the square:
So this is the equation of the shifted graph. Choice D.