To determine the best form of a quadratic equation for each of the listed attributes, consider the following:
- Axis of Symmetry: This attribute can be directly observed from the vertex form of a quadratic equation. The axis of symmetry is [tex]\(x = h\)[/tex] where [tex]\(h\)[/tex] is the horizontal shift. Therefore, the appropriate form is:
[tex]\[
f(x) = a(x-h)^2 + k
\][/tex]
- Vertex: The vertex of a parabola is easily identified in the vertex form, which is [tex]\((h, k)\)[/tex]. Thus, the best form to determine the vertex is:
[tex]\[
f(x) = a(x-h)^2 + k
\][/tex]
- x-Intercepts, or Roots: The intercept form, which factors the quadratic equation, provides the roots directly as [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex]. Thus, the most suitable form is:
[tex]\[
f(x) = a(x-r_1)(x-r_2)
\][/tex]
- y-Intercept: The standard form of a quadratic equation allows us to directly observe the y-intercept as [tex]\(c\)[/tex], where the equation crosses the y-axis when [tex]\(x = 0\)[/tex]. Hence, the best form is:
[tex]\[
f(x) = ax^2 + bx + c, a \neq 0
\][/tex]
Filling in the table, we get:
\begin{tabular}{|c|l|}
\hline
Attribute & Form \\
\hline
Axis of Symmetry & [tex]\(f(x) = a(x-h)^2 + k\)[/tex] \\
\hline
Vertex & [tex]\(f(x) = a(x-h)^2 + k\)[/tex] \\
\hline
[tex]$x$[/tex]-Intercepts, or Roots & [tex]\(f(x) = a(x-r_1)(x-r_2)\)[/tex] \\
\hline
[tex]$y$[/tex]-Intercept & [tex]\(f(x) = ax^2 + bx + c, a \neq 0\)[/tex] \\
\hline
\end{tabular}
