3.1 Quadratic Functions and Models
Ideas in this lesson
- Complete the square and interpret vertex form
- Read complete information from a parabola graph
- Construct quadratic graphs with exact intercepts
- Compare numerical and exact quadratic extrema
- Recover a quadratic from a vertex and point
- Optimize functions quadratic in a power
- Analyze projectile height and horizontal-path models
- Optimize revenue and profit with units
- Optimize a quadratic on its stated physical interval
- Optimize discrete agriculture and density models
- Optimize fenced rectangles and subdivided pens
- Optimize rectangular gutter cross-sections
- Optimize wire partitions and Norman-window areas
- Optimize demand, capacity and margin-based price models
- Prove the relation between zeros and the vertex
Complete the square and interpret vertex form · Worked example
Complete the square for \(f(x)=\dfrac{x^{2}}{2} + x - \dfrac{9}{2}\). State its vertex, axis and whether it opens upward or downward. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(f(x)=\dfrac{\left(x + 1\right)^{2}}{2} - 5\); vertex \((-1,-5)\); axis \(x=-1\); upward.
\(f(x)=\dfrac{x^{2}}{2} + x - \dfrac{9}{2}\).
Factor the leading coefficient from the variable terms; add \(1\) inside the square and compensate outside.
\(f(x)=\dfrac{1}{2}\left(x^{2} + 2 x+1\right)-5\).
Completing the square gives \(f(x)=\dfrac{\left(x + 1\right)^{2}}{2} - 5\).
The vertex is \((-1,-5)\); the sign of \(\dfrac{1}{2}\) determines the opening.
Written-work review:
Factor the leading coefficient from the variable terms; add \(1\) inside the square and compensate outside.
\(f(x)=\dfrac{1}{2}\left(x^{2} + 2 x+1\right)-5\).
Completing the square gives \(f(x)=\dfrac{\left(x + 1\right)^{2}}{2} - 5\).
The vertex is \((-1,-5)\); the sign of \(\dfrac{1}{2}\) determines the opening.
Written-work review:
- Show the requested algebra or numerical evidence, not merely a selected option.
- Include all requested coordinates, domains, units, comparisons and conclusions.
- For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.