- Factor out the leading coefficient and complete the square for \(g(x) = -x^2 + 16x - 44\).
- Rewrite \(g(x)\) in vertex form: \(g(x) = -(x - 8)^2 + 20\).
- Identify the transformations by comparing to \(f(x) = x^2\): reflection over the x-axis, horizontal shift of 8 units right, and vertical shift of 20 units up.
- The transformations are: \(\boxed{\text{reflected over the x-axis, shifted 8 units to the right, shifted 20 units up}}\).
Explanation
- Analyzing the Functions
To determine the transformations applied to the graph of \(f(x) = x^2\) to obtain the graph of \(g(x) = -x^2 + 16x - 44\), we first write the function \(g(x)\) in vertex form. The vertex form of a quadratic function is given by \(g(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. This form allows us to easily identify horizontal and vertical shifts as well as reflections.
- Completing the Square
We start with \(g(x) = -x^2 + 16x - 44\). First, factor out the coefficient of the \(x^2\) term, which is -1, to group the \(x\)-terms:
\[g(x) = -(x^2 - 16x) - 44.\]
To complete the square for the expression inside the parentheses, we take half of the coefficient of \(x\), which is \(-16/2 = -8\), and square it: \((-8)^2 = 64\). We add and subtract 64 inside the parentheses:
\[g(x) = -(x^2 - 16x + 64 - 64) - 44.\]
Distribute the -1 to the -64 to move it outside the parentheses:
\[g(x) = -(x^2 - 16x + 64) + 64 - 44.\]
Simplify the expression to obtain:
\[g(x) = -(x - 8)^2 + 20.\]
- Identifying the Transformations
Now, we compare the transformed function \(g(x) = -(x - 8)^2 + 20\) to the original function \(f(x) = x^2\).
- The negative sign in front of the \((x-8)^2\) term indicates a reflection over the x-axis.
- The \((x - 8)\) term indicates a horizontal translation of 8 units to the right.
- The \(+ 20\) constant indicates a vertical translation of 20 units up.
Thus, the transformations are a reflection over the x-axis, a horizontal shift of 8 units to the right, and a vertical shift of 20 units up. Any one of these can be identified as a transformation.
Examples
In architecture or engineering, understanding transformations is key to designing structures like parabolic arches. If you have a basic arch design modeled by \(f(x) = x^2\), you can adjust its position and orientation using these shifts and reflections to perfectly fit a structure's dimensions. For example, if you need to place a foundation arc that is reflected and shifted to support a specific entryway, calculating these exact transformations ensures the design is structurally sound and aesthetically centered.