# Quadratic Functions

• Construct quadratic functions and quadratic sequences (A-CED.A.2$^\star$, F-IF.A.3, F-BF.A.1a$^\star$).

• Represent quadratic functions using recursive formulas, expressions, tables, and graphs (A-SSE.A.1$^\star$, A-CED.A.2$^\star$, F-IF.A.7a$^\star$, F-BF.A.1a$^\star$).

• Express quadratic functions in equivalent forms for different purposes; understand the relation between vertex form and the shape of the graph (A-SSE.A.1, A-SSE.B.3$^\star$, F-IF.A.7a$^\star$, F-IF.C.8, F-BF.B.3).

• Find the average rate of change of a quadratic function over a unit interval and compare rates for successive intervals (F-IF.B.6$^\star$).

• Describe properties that distinguish linear, exponential, and quadratic functions (F-LE.A.3$^\star$).

• Model with quadratic functions (A-CED.A.2$^\star$, F-IF.B.4$^\star$, F-IF.C.7a$^\star$, F-BF.A.1a$^\star$).

*The story before this unit:*

In the unit Linear Equations, Inequalities, and Systems students developed fluency with linear functions and in unit A2 they learned about simple exponential functions. Students can use graphing technology to plot a function, find intercepts and intersection points, and find a linear regression. They know exponent rules and the application of the distributive property to combining like terms or factoring out a common factor.

*The part of the story happening in this unit:*

In this unit students build and interpret quadratic functions. They work with contexts that can be modeled by quadratic functions and compare them with contexts that can be modeled linear and exponential functions. They express quadratic functions using recursive equations (e.g. $f(n + 1) = f(n) + 2n + 1$), equations in two variables (e.g. $y = x^2 + 2x + 3$), or function notation (e.g. $f(x) = x^2+2x+3$). They understand the purpose of different forms for the quadratic expression on the right hand side in the last two cases. They graph quadratic functions expressed in different forms, and construct functions expressed in factored or vertex form for a given learn how to put a function in vertex form, and see what information can be most easily obtained from it.

*The story after this unit:*

In the unit Quadratic Equations students solve quadratic equations in one variable approximately and exactly using various methods. They continue to develop facility in algebraic manipulation of quadratic expressions and equations. In unit A5, they explore complex numbers and revisit quadratic equations to solve for complex roots.

## Sections

#### Summary

Diagnose students’ ability to

• distinguish linear from exponential functions and identify if a function is neither (F-LE.A.1$^\star$);

• evaluate quadratic expressions and solve simple quadratic equations by inspection (6.EE.A.1, 6.EE.A.2c);

• solve simple quadratic equations by inspection (6.EE.B.5, 8.EE.A.2, A-REI.A.1).

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#### Summary

• Model a context with a quadratic function and interpret values of the function in context (A-CED.A.2$^\star$, F-BF.A.1a$^\star$).

• Graph a quadratic function and interpret the graph (F-IF.C.7a$^\star$).

• Find the average rate of change over a unit interval and compare rates for successive intervals (F-IF.B.6$^\star$).

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#### Summary

• Look for structure in number sequences arising from visual patterns (A-SSE.A.1$^\star$, F-IF.A.3).

• Model the patterns with quadratic functions given by recursive descriptions, expressions, or equations (A-SSE.A.1$^\star$, A-CED.A.2$^\star$, F-BF.A.1a$^\star$).

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#### Summary

• Understand that different ways of seeing a pattern give rise to different but equivalent expressions for the function arising from the pattern (A-SSE.B.3$^\star$).

• Reason through the equivalence of expressions (A-SSE.A.2).

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#### Summary

• Distinguish between tables representing linear, exponential, and quadratic functions (F-LE.A.3$^\star$).

• Distinguish between graphs of linear, exponential, and quadratic functions (F-LE.A.3$^\star$).

• Use precise language to describe properties that distinguish linear, exponential, and quadratic functions (F-LE.A.3$^\star$).

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#### Summary

• Construct a simple quadratic model (F-BF.A.1a$^\star$).

• Use the model to solve problems and make predictions (F-IF.B.4$^\star$, F-IF.C.7a$^\star$).

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#### Summary

Assess students' ability to

• write, use and interpret quadratic function that models a given context (A-CED.A.2$^\star$, F-IF.B.4$^\star$, F-BF.A.1a$^\star$);

• express quadratic functions in equivalent forms and choose an appropriate form for a given purpose (A-SSE.A.1$^\star$, A-SSE.A.2, A-SSE.A.3);

• differentiate between graphs and tables of values for linear, exponential, and quadratic functions (F-LE.A.3$^\star$);

• label features of the graph of a quadratic function with key vocabulary words (axis of symmetry, vertex, maximum, minimum, x-intercept, y-intercept) (F-IF.C.7a$^\star$).

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#### Summary

• Understand how the structure of vertex form is related to the maximum or minimum value of the function and to the vertex of its graph (A-SSE.A.1$^\star$, F-BF.B.3).

• Use vertex form to write a possible quadratic function given the maximum or minimum of the function or the vertex of its graph (A-SSE.B.3$^\star$, F-IF.C.7a$^\star$).

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#### Summary

• Understand how the structure of factored form is related to the zeros of a function and the x-intercepts of its graph (A-SSE.A.1$^\star$, F-IF.C.7a$^\star$).

• Select the best form for expressing a quadratic function to illuminate specific features of graphs (A-SSE.B.3$^\star$, F-IF.C.8).

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#### Summary

• Fit functions to verbal descriptions and graphs using key features (F-IF.C.7a$^\star$, F-BF.B.3).

• Solve modeling problems (F-IF.B.4$^\star$, F-LE.A.1, F-LE.A.2).

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#### Summary

Select find a function that matches a given graph (F-IF.B.4$^\star$, F-LE.A.3).

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#### Summary

Assess students' ability to

• generate functions given graphs and graphs given functions (A-CED.A.2$^\star$, F-IF.C.7a$^\star$, F-IF.C.8, F-BF.A.1a$^\star$, F-BF.B.3);

• interpret expressions for quadratic functions in terms of a context it represents (A-SSE.A.1$^\star$);

• express quadratic functions in different forms for different purposes (F-IF.C.8);

• solve modeling problems using quadratic functions (F-IF.B.4$^\star$, F-BF.A.1a$^\star$).

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