Ashley Swift February 15, 2001 UNIT PLAN: EXPLORING QUADRATIC FUNCTIONS UNIT OBJECTIVES: This unit will help students learn to: graph quadratic equations with the use of technology. find the axis of symmetry and the coordinates of the vertex of a parabola. find the roots of quadratic equations. use established patterns to graph quadratic equations by hand. solve quadratic equations with the quadratic formula. use the discriminate to determine the number of real roots. SOL S SATISFIED WITH COMPLETION OF THIS UNIT: A.14 The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. A.15 The student will determine the domain and range of a relation given a graph or a set of ordered pairs and will identify the relations that are functions. A.16 The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph.
GRAPHING QUADRATIC EQUATIONS WITH THE TI-83 OBJECTIVE: Students will explore the concepts of upper and lower and the vertex. They will be able to use the trace button and the table on the TI-83 to answer questions about a parabola. OPENING ACTIVITY: The teacher and the students will throw a ball overhand and underhand around the classroom in the shape of a parabola. LESSON: While throwing a ball around the classroom the teacher will describe what a parabola is. The discussion will progress to the description of the upper and lower bound and the vertex of the equation. Students will be given a TI-83 and taught to graph a parabola and how to identify the vertex using the trace button and the table button. ASSESSMENT: The students will practice their new talent of graphing parabolas and identifying the coordinates of the vertex on the following worksheet. CLOSING ACTIVITY: There will be a classroom discussion about famous parabolas (for example: The Gateway Arch, setting off fireworks, throwing a football).
NAME: GRAPHING QUADRATIC FUNCTIONS WITH THE TI-83 Directions: Graph each of the following quadratic functions with your calculator and sketch them on this worksheet in the space provided. Please make sure you identify the coordinates of the vertex and label it an upper or lower bound. 1) x² + 16x + 59 2) 12 x² + 18x + 10 3) x² - 10x + 25 4) -2 x² -8x 1 5) 2(x-10)² + 14 6) -.5 x² -2x + 3
PARENT GRAPHS AND THEIR FAMILIES OBJECTIVES: Using a TI-83 to discover the characteristics of families of parabolas. MATERIALS: Colored Pencils and graph paper. OPENING ACTIVITY: The lesson will start off by using two student volunteers to play toss in front of the blackboard. They should toss the ball overhand and underhand with different heights and lengths. As they play this game of catch the teacher should record the shapes of the parabolas in on the blackboard. LESSON: The teacher will explain that each of the parabolas on the board are related and they are all children of the parent graph: y= x². After introduction of the parent graph y=x² the students will use a small group exploratory activity to discover what makes a parabola jump, slide, warp, and flip. The exploratory activity will require colored pencils and graph paper. The student will use the TI-83 to graph the following columns of sets of
equations. They will use a different colored pencil to graph each equation in the set on the same graph for easy comparison. Y = x² Y = x² Y = x² Y= (x-2) ² Y=.5x² Y=x²-5 Y=(x+3) ² Y=.3x² Y=x²+3 Y=(x+1) ² Y= 2x² Y= x²+.5 CLOSING ACTIVITY: Discuss what happens to the parent graph when you multiply, add, and subtract to the leading coefficient and the entire equation. ASSESSMENT: Worksheet attached
CHARACTERISTICS OF QUADRATIC FUNCTIONS OBJECTIVES: To graph the quadratic function by hand and identify axis of symmetry, coordinates of the vertex, and the zero s of the function. OPENING ACTIVITY: Introduction of the Gateway Arch of the Jefferson National Expansion Memorial in St. Louis, Missouri (pictures and dates?). It is shaped like an upside down U (parabola) but is actually a catenary. The approximate shape of the arch can be graphed by using the equation: Y = -0.00635x² + 4.0005x 0.07875 LESSON: The teacher will introduce the formal definition of the quadratic function: A quadratic function is a function that can be described by and equation of the form: Y = ax²+bx+c, where a0 The teacher will introduce the idea about the axis of symmetry by first talking about symmetry and then folding a large parabola in half and coming up the equation of that line. The teacher can then give the equation of the axis of symmetry: For the graph: y = ax²+bx+c, where a0 the axis of symmetry is x= -b/2a
The discussion should include the fact that with the x value of the vertex of the parabola given by the equation of the axis of symmetry can be used to find the exact value of the coordinates of the vertex. The class can use the example of the St. Louis Gateway Arch to find the axis of symmetry, the coordinates of the vertex, and the zero s of the function. The equation of the Gateway Arch is approximately: F(x) = -.00635x² + 4.0005x -.07875 The zero s can be found using the graphing calculator (zero and 630) CLOSING ACTIVITY: End the discussion with the determining of the distance of the base of the Arch. (630 feet) ASSESSMENT: The worksheet attached will assess the students understanding of the vertex and the axis of symmetry of a parabola.
NAME: CHARACERISTICS OF QUADRATIC FUNCTIONS Please write a brief description of how you can use the axis of symmetry to help them graph a parabola. Graph the following equations; find the equation of the axis of symmetry and the coordinates of the vertex. 1.) y = 2x² + 12x 11 2.) y = x² + 2x + 18 3.) y = 5 + 16 2x² 4.) y = -(x-2) ² + 1
QUATRATIC FORMULA OBJECTIVES: For students to understand where the quadratic formula came from, why it is useful, and how to use it. LESSON: The teacher will begin with reasons why the quadratic formula should be used (when you can not factor a quadratic function). The students will then work in pairs and take turns using the quadratic formula to solve an equation and using estimation with the TI-83. The students will then compare their solutions. For the closing discussion questions the teacher may want to discuss why the students get two solutions when using the quadratic equation, why the students may not get two real solutions, and the possibility of getting just one solution. ASSESSMENT: The following worksheet will assess the students understanding of the quadratic formula.
NAME: QUADRATIC EQUATION Use the quadratic formula to determine the solution to the following equation. The Quadratic Formula: 1.) 5y² - y- 4 = 0 2.) 24x² -14x = 6 3.) x² +6x = 36 +6x 4.) a² - 3/5a + 2/25 = 0 5.) 2w² = - (7w +3) 6.)1.34a² - 1.1a = -1.02
THE DISCRIMINANT OBJECTIVE: The students will use the discriminant to determine the number of real solutions. LESSON: The teacher will let the students know there is a quick check to find out the number of real solutions they will get when they use the quadratic equation. Next the teacher will introduce the discriminant: y = b² - 4ac, and tell the students where it came from. After a discussion with the students about what happens under the square root sign (from the quadratic equation) when they are taking the root of a negative, a perfect square, and not a perfect square, the students and the teacher will fill out a table similar to the following: When the discriminant equals: A perfect square Not a perfect square Not a perfect square, but the expression is not an integer Zero The solution is: Two integers (can be factored) Two irrational numbers Two non-integer rational solutions One integral solution ASSESSMENT: The worksheet attached will assess the students understanding of the discriminant.
NAME: THE DISCRIMINANT Use the discriminant: b² - 4ac to determine the number of solutions and the type of solutions you would get if you used the quadratic formula. Remember: When the discriminant equals: A perfect square Not a perfect square Not a perfect square, but the expression is not an integer Zero The solution is: Two integers (can be factored) Two irrational numbers Two non-integer rational solutions One integral solution EQUATION x² - 4x +1 = 0 x² + 6x + 11 = 0 x² - 4x +4 = 0 Value of the discriminant Number of x-intercepts Number of y-intercepts
NAME: TEST Write the equation of the axis of symmetry and find the coordinates of the vertex of the graph of each equation. 1.) y = -3x² + 4 2.)y = x² - 3x 4 Solve each equation by graphing (you may use your calculator). If integral roots can not be found, state the consecutive integers between which the roots lie. 3.) x² -10x + 21=0 4.) y² + 4y 3 = 0 Solve the following equations by using the quadratic formula: 5.) 4p² + 4p 15 = 0 6.) 9k² - 13k + 4 = 0 Use the discriminate to determine the number of rational roots: 7.) 9a² + 30a + 25 = 0
Choose the letter that best matches each equation or phrase: 8.) f(x) = ax² +bx + c a) equation of the axis of symmetry 9.) a geometric property of parabolas b)parabola 10.) c)quadratic formula 11.) maximum or minimum point of a parabola d)quadratic function 12.) x = -b/2a e) vertex 13.) The graph of the quadratic function f) symmetry GRAPH: be sure to label the coordinates of the vertex and the roots. 14.) x² - x - 6 = 0