Question

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).

Explanation:

Step1: Rewrite the function

Rewrite \(f(x)=-x^{2}+2x\) in vertex form.

$$
LATEXBLOCK0
$$

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.