A-REI.4.11

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.

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Formative Assessments
Formative Assessments
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Students are asked to find the solution(s) of the equation f(x) = g(x) given the graphs of f and g and explain their reasoning.
Graphs and Solutions – 2
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Students are asked to explain why the x-coordinate of the intersection of two functions, f and g, is a solution of the equation f(x) = g(x).
Graphs and Solutions -1
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Students are asked to find solutions of the equation f(x) = g(x) for two given functions, f and g, by constructing a table of values.
Using Tables
Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).
Equation Grapher
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Assessment Questions
Assessment Questions
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Videos of Expert Instruction
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Hook/Performance Tasks
Hook/Performance Tasks
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In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).
Population and Food Supply
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Students add the monthly payments together with the initial down payment and compare it to the single annual payment.
Is It Cheaper To Pay Monthly or Annually?
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This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.
Two Squares are Equal
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Students will have to decompose 450 and account for the nine rolls.
How Can You Win Every Prize At Chuck E. Cheese’s?
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This context makes for an interesting application of calculating interest.
How Much Did Peterson Lose By Not Cashing His Check?
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Students should be able to reach the conclusion that the toilets’ water usage would be about the same when the dual flush toilet is used slightly less than 50% of the time for “solids” and slightly more than 50% of the time for “liquids”.
Which Toilet Uses Less Water?