Example 2
Completion requirements
| Example 2 |
The zeros of a quadratic function are \(–2\) and \(–4\). A point on the graph of the same quadratic function is \((–5, –9)\).
Determine the equation of the quadratic function in standard form.
Determine the equation of the quadratic function in standard form.
Step 1:
Organize the information that is given.
The zeros of the function are \(–2\) and \(–4\). So \((–2, 0)\) and \((–4, 0)\) are the \(x\)-intercepts of the graph of the function.
\((–5, –9)\) is a point on the graph of the quadratic function; therefore, when \(x = –5, y = –9\).
Step 2: Select a form that works with the given information.
Because the zeros are known, a factored form of a quadratic function would be a good form to work with. Since the zeros are \(–2\) and \(–4\), two factors are \((x + 2)\) and \((x + 4)\). However, it is possible that there is also a GCF. Therefore, the function can be written as \(f\left( x \right) = a\left( {x + 2} \right)\left( {x + 4} \right)\) where \(a\) corresponds to the GCF.
Step 3: Determine the value of \(a\).
Substitute the given point, \((–5, –9)\), into the equation of the function, and solve for \(a\).
\(\begin{align}
f\left( x \right) &= a\left( {x + 2} \right)\left( {x + 4} \right) \\
-9 &= a\left( { -5 + 2} \right)\left( { -5 + 4} \right) \\
-9 &= a\left( { -3} \right)\left( { -1} \right) \\
-9 &= 3a \\
-3 &= a \\
\end{align}\)
Step 4: Write the equation of the quadratic function in factored form.
\(f\left( x \right) = -3\left( {x + 2} \right)\left( {x + 4} \right)\)
Step 5: Write the equation of the quadratic function in standard form by expanding the factored form.
\(\begin{array}{l}
f\left( x \right) = -3\left( {x + 2} \right)\left( {x + 4} \right) \\
f\left( x \right) = -3\left( {x^2 + 2x + 4x + 8} \right) \\
f\left( x \right) = -3\left( {x^2 + 6x + 8} \right) \\
f\left( x \right) = -3x^2 - 18x - 24 \\
\end{array}\)
The zeros of the function are \(–2\) and \(–4\). So \((–2, 0)\) and \((–4, 0)\) are the \(x\)-intercepts of the graph of the function.
\((–5, –9)\) is a point on the graph of the quadratic function; therefore, when \(x = –5, y = –9\).
Step 2: Select a form that works with the given information.
Because the zeros are known, a factored form of a quadratic function would be a good form to work with. Since the zeros are \(–2\) and \(–4\), two factors are \((x + 2)\) and \((x + 4)\). However, it is possible that there is also a GCF. Therefore, the function can be written as \(f\left( x \right) = a\left( {x + 2} \right)\left( {x + 4} \right)\) where \(a\) corresponds to the GCF.
Step 3: Determine the value of \(a\).
Substitute the given point, \((–5, –9)\), into the equation of the function, and solve for \(a\).
\(\begin{align}
f\left( x \right) &= a\left( {x + 2} \right)\left( {x + 4} \right) \\
-9 &= a\left( { -5 + 2} \right)\left( { -5 + 4} \right) \\
-9 &= a\left( { -3} \right)\left( { -1} \right) \\
-9 &= 3a \\
-3 &= a \\
\end{align}\)
Step 4: Write the equation of the quadratic function in factored form.
\(f\left( x \right) = -3\left( {x + 2} \right)\left( {x + 4} \right)\)
Step 5: Write the equation of the quadratic function in standard form by expanding the factored form.
\(\begin{array}{l}
f\left( x \right) = -3\left( {x + 2} \right)\left( {x + 4} \right) \\
f\left( x \right) = -3\left( {x^2 + 2x + 4x + 8} \right) \\
f\left( x \right) = -3\left( {x^2 + 6x + 8} \right) \\
f\left( x \right) = -3x^2 - 18x - 24 \\
\end{array}\)