Understand Quadratic Functions

Common Core Says...

Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student?s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a?+?b)(x?+?y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a?+?b?+?c)(x?+?y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

- Common Core Mathematics

Big Picture Lesson Planning for the Common Core

 

Week #25 Quadratics as Functions

• Understand what the parts of a quadratic mean (A.SSE.1.a)
• Understand function, domain and range in terms of a quadratic function (F.IF.1)
• Evaluate quadratic functions given a value of x in function notation (F.IF.2)
• Write an equation for a quadratic function given a relationship (F.BF.1.a)

Week #26 Graphing Quadratics

• Graph and create tables of quadratic functions and label key features (A.REI.10, F.IF.7.a & F.IF.4)
• Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (F.LE.3)
• Understand how to translate a quadratic function in a table, an equation or on a graph (F.BF.3)

Week #27 Introduction to Factoring

• Factor simple quadratics (This concept is completed in Chapter 12.) (F.IF.8.a)
• Understand the concept and implications of a zero (F.IF.8.a)
• Compare two quadratic functions to make decisions (F.IF.9)

 

Lesson Ideas

All of these resources and more can be found in the livebinder below.

Common Core Standards

A.SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.2 Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* (Emphasize quadratic, linear, and exponential functions and comparisons among them)

F.IF.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

F.IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.

F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, give a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.BF.1.a Write a function that describes a relationship between two quantities. (Emphasize linear, quadratic, and exponential functions). Determine an explicit expression, a recursive process, or steps for calculation from a context. Video explaination

F.BF.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Video explanation

F.BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x + 1)/(x-1) for x ? 1.

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