Objective
Identify solutions to a system of a quadratic function and a linear function graphically and algebraically.
Common Core Standards
Core Standards
The core standards covered in this lesson
A.REI.A.1— Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Reasoning with Equations and Inequalities
A.REI.A.1— Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.C.7— Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
Reasoning with Equations and Inequalities
A.REI.C.7— Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
A.REI.D.11— Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Reasoning with Equations and Inequalities
A.REI.D.11— Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Foundational Standards
The foundational standards covered in this lesson
A.REI.D.10
Reasoning with Equations and Inequalities
A.REI.D.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).
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Identify solutions to a system of functions graphically by noticing the number of times the functions cross.
- Use the principal that a solution to a system of functions is when the functions are equal to each other to solve a system of functions.
- Identify extraneous solutions by plugging a value back into the equation and noticing that the equation does not make a true statement.
- Use the vocabulary term “tangent line” to describe the line that intersects a parabola in only one point.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
This standard is also taught in Algebra 1. Because of the importance for AP Calculus, a review is included in this unit, which supports the next lesson.
Fishtank Plus
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Problem 1
How many solutions are in each of the following systems?
 |  |  |
$${{{f(x)={1\over2}x^2-2x+4}}}$$ $${g(x)={1\over2}x-1}$$ | $${{{f(x)={1\over2}x^2-2x+4}}}$$ $${r(x)=-x+8}$$ | $${{{f(x)={1\over2}x^2-2x+4}}}$$ $${t(x)=2}$$ |
Verify the number of solutions algebraically.
Guiding Questions
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Problem 2
The figure shows graphs of a linear and a quadratic function.
The equation that models the quadratic function is $${ y=-(x+2)^2+17}$$.
- What are the coordinates of point Q?
- What are the coordinates of point P?
Guiding Questions
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References
Illustrative Mathematics A Linear and Quadratic System
A Linear and Quadratic System, accessed on Aug. 18, 2017, 4:11 p.m., is licensed by Illustrative Mathematics under either theCC BY 4.0orCC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Calculate the solutions to the following system algebraically. Identify any extraneous solutions.
$${f(x)=x^2-2x+3}$$
$${g(x)=-x+5}$$
Additional Practice
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
- Include problems where students need to find the solution on their graphing calculators and then verify the solution algebraically.
- Include problems where students need to identify the number of solutions as well as the exact solutions of the system of functions.
- Include problems where students are given a quadratic and linear system that has two solutions and they are asked to transform the quadratic such that there are one solution and no solutions to the system.
- Include problems where students need to find the solution to a system of two quadratic equations. Do this graphically and algebraically.
- Inside Mathematics Performance Assessment Tasks Grades 3-High School Performance Assessment Tasks—high school algebra: quadratic (2009)
Lesson 10
