Unit 5-Quadratic Functions
In this unit you will study a variety of ways to solve quadratic functions and apply your learning to
analyzing real world problems.
Enduring Understandings
In Unit 5, students are provided with a family of functions whose rate of change is not constant but whose behavior is none-the-less predictable and seen often in Advanced Placement mathematics. This unit helps prepare students for Advanced Placement courses by
• Modeling motion using quadratic relationships.
• Writing equations from verbal descriptions and physical models.
• Making connections between multiple ways to represent mathematical information: numerically, graphically, verbally and algebraically.
• Increasing student ability to solve a wide-variety of equations and to choose the most appropriate solution method when needed.
• Using technology as a tool for discovering relationships, testing conjectures, and solving problems.
Essential Questions
How are quadratic functions used to model, analyze and interpret mathematical relationships?
Why is it advantageous to know a variety of ways to solve and graph quadratic functions?
Wild Words (Vocabulary)
parabola
parent function
quadratic formula
quadratic function
real roots of an equation
transformation
vertex of a parabola
Standards aligned to Springboard Unit 5
A.REI.4a The learner can solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.Description: I can identify a quadratic expression.
I can identify a perfect square trinomial by first noticing if a andc are perfect squares and if
b = 2ac.
I can factor a perfect-square trinomial.
I can complete the square of ax2+ bx + c to write the quadratic in the form of (x - p)2= q.
I can derive the quadratic formula by completing the square of ax2 + bx + c.
A.REI.4b The learner can solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for (X)2 = 49, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for the real numbers a and b.
Description: I can determine the best method to solve a quadratic equation in one variable.
I can solve quadratic equations by inspection.
I can solve quadratic equations by finding square roots.
I can solve quadratic equations by completing the square.
I can solve quadratic equations using the quadratic formula.
I can solve quadratic equations by factoring.
I can explain that complex solutions result when the radicand is negative in the quadratic formula.
I can write a complex number solution for a quadratic equation in the form of a + bi by using
i = v-1.
A.REI.5 The learner will prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solution.
Description: I can define a system of equations.
I canshow that equivalent equations result when an equation is multiplied by the same number on both sides of the equal sign.
I can solve a system of two equations in two variables by elimination.
I can demonstrate that replacing one equation with the sum of that equation and a multiple of the other creates a system with the same solutions as the original system.
A.SSE.1 The learner can interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
Description: I can define expression, term, factor, and coefficient.
I can interpret the real-world meaning of the terms, factors, and coefficients of an expression in terms of their units.
I can group the parts of an expression differently in order to better interpret their meaning.
F.BF.1b The learner will be able to write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations.
Description: I can recall the parent functions.
I can apply transformations to equations of parent functions.
I can combine different parent functions (adding, subtracting, multiplying, and/or dividing) to write a function that describes a real-world problem.