Classroom Resources (2) |

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) 17 :

[MA2015] AL1 (9-12) 32 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 20 :

[MA2015] ALT (9-12) 4 :

[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]

[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 :

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

[MA2015] AL1 (9-12) 32 :

32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

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. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

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

[MA2015] AL2 (9-12) 4 :

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

[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] ALT (9-12) 4 :

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

[MA2015] ALT (9-12) 20 :

[MA2019] AL1-19 (9-12) 6 :

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.**

In this video lesson, students learn about the **zero product property**. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in the factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the *x*-intercepts of its graph (MP7).