Example 1
Completion requirements
| Example 1 |
Determine the following characteristics of the graph shown below, and then write the equation of the quadratic function in standard form.
You can also write the equation of a quadratic function in
standard form given the zeros and at least one other point on the graph.
Direction of opening:
Coordinates of the vertex:
Equation of the axis of symmetry:
Maximum or minimum value:
\(x\)-intercepts:
\(y\)-intercept:
Domain:
Range:
Coordinates of the vertex:
Equation of the axis of symmetry:
Maximum or minimum value:
\(x\)-intercepts:
\(y\)-intercept:
Domain:
Range:
Direction of opening:
upward, therefore \(a > 0\)
Coordinates of the vertex: \((3, 7)\)
Equation of the axis of symmetry: \(x = 3\)
Maximum or minimum value: minimum of \(7\)
\(x\)-intercepts: None
\(y\)-intercept: \(25\)
Domain: {\(x | x \in\) R}
Range: {\(y | y \ge 7\), \(y \in\) R}
Using this information, determine the equation of the function.
Start with the vertex form and substitute the values of \(p\) and \(q\).
\(\begin{array}
f\left( x \right) = a\left( {x - p} \right)^2 + q \\
f\left( x \right) = a\left( {x - 3} \right)^2 + 7
\end{array}\)
Use the \(y\)-intercept to calculate \(a\).
\(\begin{align}
25 &= a\left( {0 - 3} \right)^2 + 7 \\
18 &= 9a \\
2 &= a \\
\\
f\left( x \right) &= 2\left( {x - 3} \right)^2 + 7 \end{align}\)
This equation is in vertex form. Convert to standard form by expanding and simplifying.
\(\begin{array}{l}
f\left( x \right) = 2\left( {x - 3} \right)^2 + 7 \\
f\left( x \right) = 2\left( {x^2 - 6x + 9} \right) + 7 \\
f\left( x \right) = 2x^2 - 12x + 18 + 7 \\
f\left( x \right) = 2x^2 - 12x + 25
\end{array}\)
Coordinates of the vertex: \((3, 7)\)
Equation of the axis of symmetry: \(x = 3\)
Maximum or minimum value: minimum of \(7\)
\(x\)-intercepts: None
\(y\)-intercept: \(25\)
Domain: {\(x | x \in\) R}
Range: {\(y | y \ge 7\), \(y \in\) R}
Using this information, determine the equation of the function.
Start with the vertex form and substitute the values of \(p\) and \(q\).
\(\begin{array}
f\left( x \right) = a\left( {x - p} \right)^2 + q \\
f\left( x \right) = a\left( {x - 3} \right)^2 + 7
\end{array}\)
Use the \(y\)-intercept to calculate \(a\).
\(\begin{align}
25 &= a\left( {0 - 3} \right)^2 + 7 \\
18 &= 9a \\
2 &= a \\
\\
f\left( x \right) &= 2\left( {x - 3} \right)^2 + 7 \end{align}\)
This equation is in vertex form. Convert to standard form by expanding and simplifying.
\(\begin{array}{l}
f\left( x \right) = 2\left( {x - 3} \right)^2 + 7 \\
f\left( x \right) = 2\left( {x^2 - 6x + 9} \right) + 7 \\
f\left( x \right) = 2x^2 - 12x + 18 + 7 \\
f\left( x \right) = 2x^2 - 12x + 25
\end{array}\)