Last updated: 5/26/2015

## Math- Algebra 1 Core Map Quarter 3

Solving Quadratic Equations by Completing the Square

The Quadratic Formula and the discriminant

Analyzing Functions with successive differences

Systems of quadratic and linear functions

Find solutions to quadratic and linear functions graphically

Solve systems of Quadratic and Linear Functions algebraically

Prove solutions to Systems algebraically

Applications of Systems of Quadratic and Linear Functions Graphically

Cubic Functions

Graphing cubic functions

Transforming cubic functions

Solving cubic functions

Connecting zeros, roots and solutions found algebraically to graph

Square Root Functions

Graphing Square Root functions

Transforming Square Root Functions

Solving Square Root Equations

Function Features

Increasing/Decreasing

Positive/Negative

Relative Max/Min

Symmetries

End Behavior

Build New Functions

Quarter 3

To solve quadratic equations, the polynomial should be listed in standard form

Solving quadratic equations using: factoring, graphing, completing the square, and the quadratic

formula.

The roots, zeros and solutions to quadratics are found graphically on the x-axis

Graphing quadratic equations-determining characteristics of the graph; vertex, axis of symmetry

end behavior, domain and range

Comparing transformed quadratics to the parent function

Writing quadratic equations in vertex form

Complete incomplete trinomials

Use the discriminant to determine solutions

Understand why the discriminant defines our solutions

Sytems of linear and quadratic equations

Solutions to linear and quadratic systems are the points of intersection when the functions are graphed

Solutions to linear and quadratic systems can be found algebraically using elimination and substitution.

Solutions to linear and quadratic systems can be found using graphing calculator

Solutions to linear and quadratic systems can be proven by showing solutions hold true in each function's equation.

Cubic Functions

Graphing Cubic Functions-understanding the movement

Domain and Range of Cubics

Transformations of Cubic Functions

Solving Cubic Equations and connecting the zeros, solutions or roots to the graphed function

Determining if a point is on a given cubic function

Square Root Functions

Squares of variables

Square root function is an increasing function

Characteristics of parent function

Domain and Range of function

Graph of Square root function

Transformations of square root function

Domain and Range of square root function

Rationalizing the denominator

Simplifying nth root expressions

Function Features

Each function has specific end behavior

There are domian intervals where the function is positive or negative

Some functions have minimum or maximum values, others relative minima or maxima

Some functions are symmetrical

Why and how does the graph of a quadratic equation differ from a linear function?

What part of the function creates the symmetry that parabolas have?

How can we find the line of symmetry of a graphed parabola?

Is there more than one line of symmetry of the parabola?

How do quadratic equations display themselves in real life?

Why do quadratic equations have two roots?

How can we use a graph to solve a quadratic equation?

Is there a connection between the roots of the quadratic equation and the factors of the polynomials?

Is there a way to predict the types of roots a quadratic might have?

Can we determine the turning point and axes of symmetry of a parabola without graphing?

Why is the vertex referred to as the turning point?

How can we use our knowledge of perfect trinomials to complete the square and solve a quadratic?

How was the quadratic equation developed and how can we use it to solve quadratic equations?

What is the discriminant and what can we conclude from its value?

How should we select the method to use to solve quadratic equations?

Systems of linear and quadratic equations

What does a system of a linear and quadratic functions look like?

At most, how many shared points can a quadratic and linear function share?

What is the least amount of points, a quadratic and linear function share?

Is there a way to determine these shared points algebraically?

What does a graphed system with no solutions look like?

How can you know that the solutions you determined, are in fact points on both lines?

Can we determine the solutions using a graphing calculator?

Function Features

How can we determine the end behavior of functions as they approach infinity or negative infinity?

Can we determine domain intervals where a function is positive or negative?

Do all functions have maximum and minimum points?

Are all functions symmetrical?

Can we build new functions from given functions?

 (1) A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (1) A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (1) A-REI.4 Solve quadratic equations in one variable. (1) A-REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. (1) A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines. (1) A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (1) 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. (1) 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). (1) F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (1) 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.★ (1) F-IF.5 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.★ (1) F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (1) F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima. (1) F-IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (1) 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. (1) 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, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Write the quadratic equation in standard form

Using the zero product property to find the zeros and the roots

Determine the axis of symmetry on the graph

Determine the axis of symmetry from a chart

Determine the axis of symmetery using the formula

Determine the maximum and minimum (vertex)

Determine the domain and range of quadratic functions

Determine the transformations of a quadratic function from the vertex form of parabola

Determine the vertex, from vertex form of a parabola

Write a quadratic function in vertex form

Determine the interval of the graph that is increasing or decreasing

Examine end behavior of the quadratic function

Determine the x and y intercepts graphically and algebraically

Rate of change of a quadratic function

Determine from a graph if there is one root, two distinct roots or no real roots

Determine the roots by completing the square

Finding the maximum or minimum using only algebra

Simplifying roots

Systems of linear and quadratic systems

Graph systems of linear and quadratic functions

Determine the points of intersection are the solutions

Prove the solution point is on both graphed functions

Find the systems of linear and quadratic systems by using algebraic methods

Cubic Functions

Graphing a cubic function from a table

Graphing a cubic function from list using graphing calculator

Transforming the cubic function

Solving a cubic function by factoring

Connecting the zeros, roots and solutions to the x-intercepts

Determining if a point is on the give cubic function

Square Root Functions

Graph the square root function

Determine the domain and range of the parent function

Determine the transformations of a square root function when compared to the parent

Explain why the parent function has only domain values greater than zero

Rationalize the denominator

Solve square root equations

Determine extraneous roots

Function Features

Determine end behavior

Determine domain intervals where a function is increasing or decreasing

Determine the maximum and minimum of a function

Determine if a function is symmetrical

Build a new function from a given function

axis of symmetry

vertex

minimum

maximum

relative minima

relative maxima

double root

translations

transformation

dilation

vertex form

completing the square

discriminant

successive differences

parent function

stretch

shrink

factors

zero property

roots

zeros

completed square

systems

cubic root

square root function

exraneous roots

rationalize

domain interval

end behavior

Pass the paper

Pair share

Exit Notes

Classroom Assessments

Writing prompt

Dailly spiral review

Weekly spiral review

Benchmarks

Self-Correcting Lab Assessment

Algebra I Core Assessment

STARS Assessment

Algebra 1 - McGraw Hill Education

Algebra 1- Amsco School Publications

Big Ideas Algebra 1- Big Ideas Learning

NYS Modules