Example 1
Completion requirements
| Example 1 |
Sketch \(f\left( x \right) = x^2 + 6x + 5 \) without using the graphing feature on your calculator. Verify using your calculator.
Step 1: Write down what you know, given the standard form.
Factoring gives
\(\begin{array}{l}
f\left( x \right) = x^2 + 6x + 5 \\
f\left( x \right) = \left( {x + 5} \right)\left( {x + 1} \right) \\
\end{array}\)
\(x\)-intercept(s): \(–5\) and \(–1\)
\(y\)-intercept: \(5\), since \(c = 5\)
Direction of opening: upward because \(a > 0\)
Step 2: Convert the quadratic function into vertex form by completing the square. Then, write what you know, given the vertex form.
\(\begin{array}{l}
f\left( x \right) = x^2 + 6x + 5 \\
f\left( x \right) = \left( {x^2 + 6x} \right) + 5 \\
f\left( x \right) = \left( {x^2 + 6x + 3^2 - 3^2 } \right) + 5 \\
f\left( x \right) = \left( {x^2 + 6x + 3^2 } \right) + 5 - 9 \\
f\left( x \right) = \left( {x + 3} \right)^2 - 4 \\
\end{array}\)
Vertex: \((-3, -4)\)
Domain: {\(x | x \in \) R}
Range: {\(y | y \ge -4\), \(y \in\) R}
Minimum: \(-4\)
Axis of symmetry: \(x = -3\)
Step 3: Graph the points you know. Use the axis of symmetry to determine more points until at least 5 are plotted. Connect the points using a smooth curve. Label the function.
Note that the point \((–6, 5)\) can be found using the axis of symmetry and the \(y\)-intercept. The \(y\)-intercept is \(3\) units to the right of the axis of symmetry. As such, a point on the graph with the same \(y\)-coordinate will exist \(3\) units to the left of the axis of symmetry.
Step 4: Verify using your calculator. Use the same window settings as you used in your sketch.
Factoring gives
\(\begin{array}{l}
f\left( x \right) = x^2 + 6x + 5 \\
f\left( x \right) = \left( {x + 5} \right)\left( {x + 1} \right) \\
\end{array}\)
\(x\)-intercept(s): \(–5\) and \(–1\)
\(y\)-intercept: \(5\), since \(c = 5\)
Direction of opening: upward because \(a > 0\)
Step 2: Convert the quadratic function into vertex form by completing the square. Then, write what you know, given the vertex form.
\(\begin{array}{l}
f\left( x \right) = x^2 + 6x + 5 \\
f\left( x \right) = \left( {x^2 + 6x} \right) + 5 \\
f\left( x \right) = \left( {x^2 + 6x + 3^2 - 3^2 } \right) + 5 \\
f\left( x \right) = \left( {x^2 + 6x + 3^2 } \right) + 5 - 9 \\
f\left( x \right) = \left( {x + 3} \right)^2 - 4 \\
\end{array}\)
Vertex: \((-3, -4)\)
Domain: {\(x | x \in \) R}
Range: {\(y | y \ge -4\), \(y \in\) R}
Minimum: \(-4\)
Axis of symmetry: \(x = -3\)
Step 3: Graph the points you know. Use the axis of symmetry to determine more points until at least 5 are plotted. Connect the points using a smooth curve. Label the function.
Note that the point \((–6, 5)\) can be found using the axis of symmetry and the \(y\)-intercept. The \(y\)-intercept is \(3\) units to the right of the axis of symmetry. As such, a point on the graph with the same \(y\)-coordinate will exist \(3\) units to the left of the axis of symmetry.
Step 4: Verify using your calculator. Use the same window settings as you used in your sketch.