Question

graph the function (f(x)= -x^{2}+4x) by starting with the graph of (y = x^{2}) and using transformations (shifting, stretching/compressing, and/or reflecting). use the graphing tool to graph the function.

Explanation:

Step1: Rewrite the function in vertex - form

First, complete the square for \(f(x)=-x^{2}+4x\).
\[

$$\begin{align*} f(x)&=-(x^{2}-4x)\\ &=-(x^{2}-4x + 4-4)\\ &=-( (x - 2)^{2}-4)\\ &=-(x - 2)^{2}+4 \end{align*}$$

\]

Step2: Identify the base - function and transformations

The base - function is \(y = x^{2}\).

1. Reflection: The negative sign in front of \((x - 2)^{2}\) reflects the graph of \(y=x^{2}\) about the \(x\) - axis.
2. Horizontal shift: The \((x - 2)\) term shifts the graph of \(y = -x^{2}\) 2 units to the right.
3. Vertical shift: The \(+4\) term shifts the graph of \(y=-(x - 2)^{2}\) 4 units up.

To graph the function:

1. Start with the graph of \(y = x^{2}\), which is a parabola opening upwards with vertex at the origin \((0,0)\).
2. Reflect it about the \(x\) - axis to get \(y=-x^{2}\), a parabola opening downwards with vertex at \((0,0)\).
3. Shift the graph of \(y = -x^{2}\) 2 units to the right to get \(y=-(x - 2)^{2}\), with vertex at \((2,0)\).
4. Shift the graph of \(y=-(x - 2)^{2}\) 4 units up to get \(y=-(x - 2)^{2}+4\), with vertex at \((2,4)\).

Answer:

Graph the parabola \(y = -x^{2}+4x\) (or \(y=-(x - 2)^{2}+4\)) which is a parabola opening downwards with vertex at \((2,4)\) obtained by reflecting \(y = x^{2}\) about the \(x\) - axis, shifting 2 units to the right and 4 units up.