Classroom Resources (4) |

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 16 :

[MA2015] AL1 (9-12) 17 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 13 :

[MA2015] AL2 (9-12) 20 :

[MA2015] AL2 (9-12) 24 :

[MA2015] AL2 (9-12) 29 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

[MA2015] ALT (9-12) 20 :

[MA2015] ALT (9-12) 24 :

[MA2015] ALT (9-12) 29 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 6 :

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

16 ) 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-REI1]

[MA2015] AL1 (9-12) 17 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] AL2 (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] AL2 (9-12) 24 :

24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]

[MA2015] AL2 (9-12) 29 :

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

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.

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] ALT (9-12) 24 :

24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]

[MA2015] ALT (9-12) 29 :

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

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.

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

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. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

[MA2019] AL1-19 (9-12) 11 : 11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. **Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.**

[MA2019] AL1-19 (9-12) 15 : 15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. *Note: 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*.*

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Limit to linear, quadratic, exponential, and absolute value functions.**

In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).

The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 16 :

[MA2015] AL1 (9-12) 17 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 21 :

[MA2015] AL2 (9-12) 27 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

[MA2015] ALT (9-12) 21 :

[MA2015] ALT (9-12) 27 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 9 :

16 ) 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-REI1]

[MA2015] AL1 (9-12) 17 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] AL2 (9-12) 27 :

27 ) 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.* [A-REI11]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] ALT (9-12) 27 :

27 ) 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.* [A-REI11]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

9. Select an appropriate method to solve a quadratic equation in one variable.

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. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

[MA2019] AL1-19 (9-12) 10 : 10. Select an appropriate method to solve a system of two linear equations in two variables.

a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

[MA2019] AL1-19 (9-12) 12 : 12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. **Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.**

[MA2019] AL1-19 (9-12) 14 : 14. Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane.

*Note: The graph of a relation often forms a curve (which could be a line).*

[MA2019] AL1-19 (9-12) 15 : 15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. *Note: 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*.*

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Limit to linear, quadratic, exponential, and absolute value functions.**

[MA2019] AL1-19 (9-12) 19 : 19. 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).*

a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.

*Note: Include cases where *f(x)* is a linear, quadratic, exponential, or absolute value function and *g(x) *is constant or linear.*

[MA2019] AL1-19 (9-12) 20 : 20. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.

This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation *y* = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 16 :

[MA2019] AL1-19 (9-12) 23 :

16 ) 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-REI1]

[MA2019] AL1-19 (9-12) 23 :

23. Identify the effect on the graph of replacing *f(x) by f(x)+k,k·f(x), f(k·x),* 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 explain the effects on the graph, using technology as appropriate. **Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.**

[MA2019] AL1-19 (9-12) 30 : 30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

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

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.

In Module 3, Topic C, students extend their understanding of piecewise functions and their graphs including the absolute value and step functions. They learn a graphical approach to circumventing complex algebraic solutions to equations in one variable, seeing them as *f*(*x*) = *g*(*x*) and recognizing that the intersection of the graphs of *f*(*x*) and *g*(*x*) are solutions to the original equation (A-REI.D.11). Students use the absolute value function and other piecewise functions to investigate transformations of functions and draw formal conclusions about the effects of a transformation on the function’s graph (F-IF.C.7, F-BF.B.3)**.**

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 15 :

[MA2015] AL1 (9-12) 16 :16 ) 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-REI1]

[MA2019] AL1-19 (9-12) 13 :

15 ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [A-CED4]

Example: Rearrange Ohm's law *V* = *IR* to highlight resistance *R*.

[MA2015] AL1 (9-12) 16 :

[MA2019] AL1-19 (9-12) 13 :

13. Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. **Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.**

Throughout middle school, students practice the process of solving linear equations (6.EE.5, 6.EE.7, 7.EE.4, 8.EE.7) and systems of linear equations (8.EE.8). Now, in Module 1, Topic C, instead of just solving equations, they formalize descriptions of what they learned before (variable, solution sets, etc.) and are able to explain, justify, and evaluate their reasoning as they strategize methods for solving linear and non-linear equations (A-REI.1, A-REI.3, A-CED.4). Students take their experience solving systems of linear equations further as they prove the validity of the addition method, learn a formal definition for the graph of an equation and use it to explain the reasoning of solving systems graphically, and graphically represent the solution to systems of linear inequalities (A-CED.3, A-REI.5, A-REI.6, A-REI.10, A-REI.12).