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Graph	Characteristic	Answer
	Direction of opening	downward, therefore \(a < 0\)
	Coordinates of the vertex	\((4, 9) \)
	Equation of the axis of symmetry	\(x = 4\)
	Maximum of minimum value	maximum of \(9 \)
	\(x\)-intercept(s)	\(1\) and \(7 \)
	\(y\)-intercept	\(–7 \)
	Domain	{\(x | x \in\) R}
	Range	{\(y | y \le 9\), \(y \in\) R}

Using this information, determine the equation of the function.

\(\begin{array}{l}
 f\left( x \right) = a\left( {x - p} \right)^2 + q \\
 f\left( x \right) = a\left( {x - 4} \right)^2 + 9 \\
 \end{array}\)

Use the \(y\)-intercept to calculate \(a\).

\(\begin{align}
  -7 &= a\left( {0 - 4} \right)^2 + 9 \\
  -16 &= 16a \\
  -1 &= a \\
  \\
 f\left( x \right) &= -\left( {x - 4} \right)^2 + 9  \end{align}\)

This equation is in vertex form. Convert to standard form by expanding and simplifying.

\(\begin{array}{l}
 f\left( x \right) = -\left( {x - 4} \right)^2  + 9 \\
 f\left( x \right) = -\left( {x^2 - 8x + 16} \right) + 9 \\
 f\left( x \right) = -x^2 + 8x - 16 + 9 \\
 f\left( x \right) = -x^2 + 8x - 7 \\
 \end{array}\)

One zero indicates that both factors are the same for the function.

\(y = a\left( {x - 3} \right)^2 \)

Use the given point to determine \(a\).

\(\begin{align}
  -16 &= a\left( {1 - 3} \right)^2  \\
  -16 &= a\left( { -2} \right)^2  \\
  -4 &= a \\
 \end{align}\)

Substitute \(–4\) for \(a\) and expand to standard form.

\(\begin{array}{l}
 y = -4\left( {x - 3} \right)^2  \\
 y = -4\left( {x^2 - 6x + 9} \right) \\
 y = -4x^2 + 24x - 36 \\
 \end{array}\)