MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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3.1 Quadratic Functions and Models

Focus on vertex and maximum value.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Complete the square and interpret vertex formWorked steps · five questions02Read complete information from a parabola graphWorked steps · five questions03Construct quadratic graphs with exact interceptsWorked steps · five questions04Compare numerical and exact quadratic extremaWorked steps · five questions05Recover a quadratic from a vertex and pointWorked steps · five questions06Optimize functions quadratic in a powerWorked steps · five questions07Analyze projectile height and horizontal-path modelsWorked steps · five questions08Optimize revenue and profit with unitsWorked steps · five questions09Optimize a quadratic on its stated physical intervalWorked steps · five questions10Optimize discrete agriculture and density modelsWorked steps · five questions11Optimize fenced rectangles and subdivided pensWorked steps · five questions12Optimize rectangular gutter cross-sectionsWorked steps · five questions13Optimize wire partitions and Norman-window areasWorked steps · five questions14Optimize demand, capacity and margin-based price modelsWorked steps · five questions15Prove the relation between zeros and the vertexWorked steps · five questions
Lesson brief and rules

3.1 Quadratic Functions and Models

Vertex form a(x−h)2+ka(x-h)^{2}+k has vertex (h,k)(h,k); the sign of a sets its opening.

Worked example 11
01 / First define a quadratic function and explain the role of its leading coefficient. Complete the square for f(x)=2x2+12x+9f(x)=2 x^{2} + 12 x + 9. 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.
02 / A quadratic has the form ax2+bx+cax^{2}+bx+c with a≠0a\ne0; its degree is exactly 22. The sign of aa controls opening, and its magnitude controls vertical scale.
03 / f(x)=2x2+12x+9f(x)=2 x^{2} + 12 x + 9.
04 / Factor the leading coefficient from the variable terms; add 99 inside the square and compensate outside.
05 / f(x)=2(x2+6x+9)−9f(x)=2\left(x^{2} + 6 x+9\right)-9.
06 / Completing the square gives f(x)=2(x+3)2−9f(x)=2 \left(x + 3\right)^{2} - 9.
07 / The vertex is (−3,−9)(-3,-9); the sign of 22 determines the opening.
Worked example 2 · Complete the square and interpret vertex form

Complete the square for f(x)=−3x2−12x−19f(x)=- 3 x^{2} - 12 x - 19. 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.

f(x)=−3x2−12x−19f(x)=- 3 x^{2} - 12 x - 19.
Factor the leading coefficient from the variable terms; add 44 inside the square and compensate outside.
f(x)=−3(x2+4x+4)−7f(x)=-3\left(x^{2} + 4 x+4\right)-7.
Completing the square gives f(x)=−3(x+2)2−7f(x)=- 3 \left(x + 2\right)^{2} - 7.
The vertex is (−2,−7)(-2,-7); the sign of −3-3 determines the opening.
Written-work review:

Read the graphs

Read complete information from a parabola graph

Read the displayed parabola. Give the vertex, both types of intercept, domain, range, and extreme value.
Graph of fAxes labeled x and y. Window x from -5 to 1, y from -12 to 12. Open circles exclude endpoints; filled circles include them.-5-4-3-2-101-12-9-6-3036912f: solidxy
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.

The turning point is (−2,−1)(-2,-1).
The graph meets the horizontal axis at −3-3 and −1-1.
At the vertical axis the output is 33.
The curve extends indefinitely; its vertex bounds the range.
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.

Answer: Vertex (−2,−1)(-2,-1); xx-intercepts (−3,0)(-3,0) and (−1,0)(-1,0); yy-intercept (0,3)(0,3). Domain: all real numbers. Range: [−1,∞)[-1,\infty). Minimum value −1-1.

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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