Intro to Quadratic Equations

Name: Eric Reyes

Title of lesson: Intro to Quadratic Equations, Part 1

Date of lesson: Friday Week 2

Length of lesson: 1 Day

Description of the class: Mathematics Name of course: Algebra

Grade level: 9

Honors or regular: Regular

Source of the lesson:

Handouts: http://www.purplemath.com/modules/grphquad2.htm

TEKS addressed:

(a) Basic understandings

(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.

(d) Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.

3) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. Following are performance descriptions.

(A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations.

(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.

I. Overview

In this lesson, students will work with quadratic equations and become familiar with their applications.

II. Performance or learner outcomes

Students will be able to:

· Recognize quadratic equations.

· Describe the graph of quadratic equations.

III. Resources, materials and supplies needed

IV. Supplementary materials, handouts.

· Handout and questions

Five-E Organization

Teacher Does Probing Questions Student Does/Responses

Engage: Ask student of any previous knowledge or understanding of quadratics.

1. Who knows the formula for quadratic equation? 2. Who knows what the graph of a quadratic looks like? Can you draw it? 3. What are the characteristics of a quadratic and its graph?

1. y= ax^x + bx + c 2. It looks like a “U” shaped curve. 3. It can open up or down, be wide or thin.

Explore: Give students the formula for a quadratic. y=ax^2+bx+c. Give the definition of a quadratic function: A quadratic is a function of the form: f(x) = ax^2 + bx + c where a is non zero. The graph of a quadratic is called a parabola. Have students then start working with the equation itself. First have students plug and graph values for the simplest for of a quadratic: y= ax^2 For now let a be a constant of their choosing. Then have students plug and graph values with the form: y= = ax^2 + bx leaving a and b as any constant. Then have students explore the entire graph = ax^2 + bx+ c with a b and c as constants.

1. Why must a be non zero? 2. What values can b and c take? 3. What makes this formula/equation unique? 4. What are some characteristics you can identify by plugging in numbers? 5. Compare and contrast the graphs of your equations. How do they relate/differ with the coefficients be added/left out?

1. To give the quadratic its curvature. If a was zero, you would have a linear function of the form bx + c. This is not quadratic. 2. Any values. If they were zero, you would have the quadratic ax^2 3. The first coefficient is followed by x^2, the second by x and the third is by itself. 4. Depending on how big or small the numbers you plug in, the graph will be different. Bigger or smaller. Students graph and draw and label pictures of the equation with different values.

Explain: Explain to students the meaning of the leading coefficient a. The general form of a quadratic is "y = ax2 + bx + c". For graphing, the leading coefficient "a" indicates how "fat" or "skinny" the parabola will be. Have students construct parabolas with different “a” values. Explanation of the vertex. Parabolas always have a lowest point (or a highest point, if the parabola is upside-down). This point, where the parabola changes direction, is called the "vertex". One point for you to remember is that, if you have an equation where a is, say, negative, and you're coming up with plot points that make it look like the quadratic is right-side-up, then you need to go back and check your work.

1. if a is greater than 1, how the graph of the parabola look like? 2. If a is less than 1, how will the graph the parabola look like? 3. What if a was negative? 4. How can we rewrite the quadratic formula in a form to locating its intercepts?

1. thinner skinny 2. fatter larger 3. then the parabola would be point downward. 4. y = ax² + bx + c y - c = ax² + bx y + b²/(4a) - c = ax² + bx + b²/(4a) y + b²/(4a) - (4ac)/(4a) = a(x² + (b/a)(x) + b²/(4a²)) y + b²/(4a) - (4ac)/(4a) = a(x + b/(2a))² y + (b² - 4ac)/(4a) = a(x + b/(2a))²

Extend / Elaborate: Have students derive a simpler form of the vertex formula.

For what values can we assighn to easily calculate the vertices of a parabola and calculate a,b,c?

h=-b/2a k= c-b^2/4a^2 y=a(x-h)^2+k

Evaluate: Graph and find the vertex of: y=3x^2+x-2 y=7x^x+3x+9 y=8x^2+2x+18

Students work on the problems and present to the class.