Lesson 8 Quadratic Quarters Practice Understanding

Learning Focus

Solve quadratic equations efficiently and accurately.

Identify information about the graph of a quadratic function from the equation.

What information do we get from each form of a quadratic equation, and which form is best for a particular purpose?

How can you determine the most efficient strategy for solving any particular quadratic equation?

Open Up the Math: Launch, Explore, Discuss

The word “quadratic” comes from the Latin word that means to make a square. Each of the following big squares are made of four smaller squares (or quarters). Some of the quarters are given, and some of the quarters need to be completed. Your quest is to find each of the missing quarters to complete the quadratic squares. Use the space on the side of the square to show your work. Problems 1–4 do not require technology, but you might need technology for problems 5–7.

1.

Function in standard form: $f (x) = x^{2} - 2 x - 8$

Function in factored form:

$x$ -intercepts:

$y$ -intercept:

Graph of the function:

2.

Function in factored form: $f (x) = (x - 7) (x + 1)$

Function in standard form:

$x$ -intercepts:

$y$ -intercept:

Line of symmetry:

Vertex:

3.

Function in standard form: $f (x) = x^{2}$

Vertex:

Line of symmetry:

Domain:

Solutions to the equation $x^{2} = 27$ :

Graph of the function:

4.

Function in standard form: $y = - x^{2} - 6 x + 16$

Vertex:

Interval(s) of increase:

Interval(s) of decrease:

Function in factored form:

$x$ -intercepts:

Line of symmetry:

5.

Equation: ${(x - 3)}^{2} = 8$

Algebraic solution method:

Solutions written with square roots:

Solutions written with decimals:

Graphical solution method:

6.

Equation: $x^{2} + x - 6 = 0$

Algebraic solution method:

Describe the relationship between the solutions to the equation and $x$ -intercepts of the graph.

Graphical solution method:

7.

Graph of the function:

Equation of the function:

$x$ -intercepts:

$y$ -intercept:

Line of symmetry:

Domain:

Range:

Interval(s) of increase:

Ready for More?

Work with three other students that have finished the task to make your own quadratic quarters. The first student starts by completing square 1. They pass the problem to the next student who completes square 2. They pass to the next student for square 3, and the last student completes square 4. If you have time, start again with a different student starting with a new function in square 1.

1: Write a quadratic function in factored form.	2: Find $x$ -intercepts, the $y$ -intercept, the vertex, and the line of symmetry.
3: Write the function in standard form.	4: Graph the function.

Takeaways

Solving Quadratic Equations:

Factoring is an efficient strategy
Taking the square root of both sides and using inverse operations is an efficient strategy when
Graphing is an efficient strategy when

Graphing Quadratic Functions:

The information that is readily available in factored form is
The information that is readily available in standard form is

Lesson Summary

In this lesson, we compared methods for solving quadratic equations. We found that some equations lend themselves to one method and other equations are more efficiently solved with other methods. Using technology to graph is always a useful way to check solutions.

Retrieval

Find the length of the missing side in each right triangle.

1.

2.

Write the equation for the quadratic function with the given $x$ -intercepts in factored form and standard form. The coefficient for the $x^{2}$ term is $1$ .

3.

$(- 5, 0)$ and $(2, 0)$

4.

$(3, 0)$ and $(11, 0)$