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Step 1: Start by factoring the function to find the zeros of the function, which corresponds to the \(x\)-intercepts of the graph of the function.

\(\begin{array}{l}
 y = -x^2 + 2x + 3 \\
 y = -x^2 - x + 3x + 3 \\
 y = -x\left( {x + 1} \right) + 3\left( {x + 1} \right) \\
 y = \left( {x + 1} \right)\left( { -x + 3} \right) \\
 \end{array}\)

\(x\)-intercepts: \(–1\) and \(3\)

\(y\)-intercept: \(3\), since \(c = 3\)

Direction of opening: downward because \(a < 0\)

Step 2: To find the vertex, domain and range, maximum, and axis of symmetry, convert the function to vertex form.

\(\begin{array}{l}
 y = -x^2 + 2x + 3 \\
 y = - \left( {x^2 - 2x} \right) + 3 \\
 y = - \left( {x^2 - 2x + 1 - 1} \right) + 3 \\
 y = - \left( {x^2 - 2x + 1} \right) + 3 + 1 \\
 y = - \left( {x - 1} \right)^2 + 4 \\
 \end{array}\)

Vertex: \((1, 4)\)
Domain: {\(x | x \in \thinspace \rm{R}\)}

Range: {\(y | y \le 4, y \in \thinspace \rm{R}\)}

Maximum: \(4\)

Axis of symmetry: \(x = 1 \)

Step 3: Plot five points (use the axis of symmetry to find any extras required), then draw a smooth curve through the points. Label the function.


Step 4: Verify using your calculator.