graph the function ( f(x)=-x^{2}+2x ) by starting with the graph of ( y = x^{2} ) and using transformations…

graph the function ( f(x)=-x^{2}+2x ) by starting with the graph of ( y = x^{2} ) and using transformations (shifting, stretching/compressing, and/or reflecting).

graph the function ( f(x)=-x^{2}+2x ) by starting with the graph of ( y = x^{2} ) and using transformations (shifting, stretching/compressing, and/or reflecting).

Answer

Explanation:

Step1: Rewrite the function

Rewrite (f(x)=-x^{2}+2x) in vertex form. [ \begin{align*} f(x)&=-(x^{2}-2x)\ &=-(x^{2}-2x + 1-1)\ &=-(x - 1)^{2}+1 \end{align*} ]

Step2: Analyze the transformations

  1. Reflection: The negative sign in (y =-(x - 1)^{2}+1) reflects the graph of (y=x^{2}) about the (x) - axis.
  2. Horizontal shift: The (x-1) term shifts the graph of (y =-x^{2}) (after reflection) 1 unit to the right.
  3. Vertical shift: The (+1) term shifts the graph of (y=-(x - 1)^{2}) (after reflection and horizontal shift) 1 unit up.

Step3: Graph the function

  • Start with the graph of (y = x^{2}), which is a parabola opening upwards with vertex ((0,0)).
  • Reflect it about the (x) - axis to get (y=-x^{2}) (a parabola opening downwards with vertex ((0,0))).
  • Shift the graph of (y=-x^{2}) 1 unit to the right to get (y=-(x - 1)^{2}) (vertex ((1,0))).
  • Shift the graph of (y=-(x - 1)^{2}) 1 unit up to get (y=-(x - 1)^{2}+1) (vertex ((1,1))).

Answer:

The graph of (y=-x^{2}+2x) is obtained by reflecting the graph of (y = x^{2}) about the (x) - axis, then shifting it 1 unit to the right and 1 unit up.