Section 6: Quadratic Equations and Functions – Part 2

1 Section 6: Quadratic Equations and Functions Part 2 Section 6 Topic 1 Observations from a Graph of a Quadratic Function Let s review some things we learned earlier about the information we can gather from the graph of a Quadratic function. Vertex: ( , ) Axis of symmetry: = -intercept(s): ( , ) and ( , ) -intercept: ( , ) Vertex: ( , ) Axis of symmetry: = -intercept(s): none -intercept: ( , ) Let s Practice! 1. The graph shows the height of a rocket from the time it was launched from the ground. Use the graph to answer the questions below. a. What is the -intercept? ( , ) b. What does the -intercept represent? The point from which the rocket was launched. c. What are the -intercepts? ( , ) and ( , ) d.

2 What do the -intercepts represent? The time the rocket was launched and the number of seconds it took for the rocket to hit the ground. e. What is the maximum height of the rocket? feet f. When will the rocket reach its maximum height? seconds g. When is the graph increasing? Set Notation < < Interval Notation , h. When is the graph decreasing? Set Notation < < Interval Notation , i. What is the domain of the graph? Set Notation Interval Notation , j. What is the range of the graph? Set Notation Interval Notation , We can also use the graph to write the equation of the Quadratic function. Recall the standard form of a Quadratic equation. = 3+ + There is another form of the Quadratic equation called vertex form.

3 Vertex Form: ( )= ( )3+ ( , ) is the vertex of the graph. determines if the graph opens up or down. also determines if the parabola is vertically compressed or stretched. To write an equation in vertex form from a graph, follow these steps: Step 1: Substitute the vertex, ( , ), and the coordinates of another point on the graph, ( , ( )), into ( )= ( )3+ . Step 2: Solve for . Step 3: Substitute ( , ) and into vertex form. 2. Recall our graph from exercise 1. a. Substitute the vertex, ( , ), and the coordinates of another point on the graph, , , into ( )= ( )3+ and solve for . , =( , ) Coordinate: ( , ) = + = = = . b. Write the function for the graph in vertex form.

4 = + or = . + Try It! 3. Consider the graph below. a. State five observations about the graph. Vertex: ( , ) -intercepts ( , ) and ( , ) -intercept ( , ) Domain: = or , Range: or , ) Models a Quadratic where is positive. b. Write the equation of the graph. = + , =( , ) Coordinate: ( , ) = + = = = + BEAT THE TEST! 1. The graph of a Quadratic function is shown below. Which statements about this graph are true? Select all that apply. The graph has a -intercept at 0,8. The graph has a relative maximum at ( 1,9). The graph has an -intercept at (2,0). The line of symmetry is the -axis. The graph has zeros of 4 and 2. The graph represents the function = 13+9.

5 Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to to learn more and get started! Section 6 Topic 2 Nature of the Zeros of Quadratic Equations and Functions Let s use the Quadratic formula to discuss the nature of the zero(s) of a Quadratic function. Consider the graph of the function = 3 4 +4. Where does the parabola intersect the -axis? At ( , ) Use the Quadratic formula to find the zero(s) of the function. = ( )( ) ( )= = = = Consider the graph of the function = 3+6 +8. Where does the parabola intersect the -axis? At ( , ) and ( , ) Use the Quadratic formula to find the zero(s) of the function.

6 = ( )( ) ( )= = = = + = = or = = = Consider the graph of the function = 3+6 11. Where does the parabola intersect the -axis? It doesn t intersect the -axis. Use the Quadratic formula to find the zero(s) of the function. = ( )( ) ( )= = no real zeros Let s Practice! 1. Use the discriminant to determine if the following Quadratic Equations have complex or real zero(s). a. 2 3 3 10=0 ( )( )= + = The equation has two real zeros. b. 3 6 +9=0 ( )( )= = The equation has one real zero. c. = 3 8 +20 ( )( )= = The equation has complex zeros. When using the Quadratic formula, if the discriminant of the Quadratic function (the part under the radical) is a negative number, then the zeros are non-real, complex solutions.

7 Try It! 2. Create a Quadratic equation that has complex zeros. Justify your answer. Answers vary. Sample answer: = , = , = = + ( )( )= = 3. Create a Quadratic equation that has one real zero. Answers vary (Any perfect square trinomial) Sample answer: = , = , = = + + ( )( )= = BEAT THE TEST! 1. Which of the following Quadratic Equations have real zeros? Select all that apply. = 3 3+5 11 = 3 12 +6 =2 3+ +6 =5 3 10 3 = 3 2 +8 Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so.

8 Go to to learn more and get started! Section 6 Topic 3 graphing Quadratics Using a Table Suppose you jump into a deep pool of water from a diving platform that is 25 feet above the ground. Your height with respect to time can be modeled by the function =25 16 3, where is time in seconds. Complete the table below. Time (seconds) 0 1 Elevation (feet) Graph function ( ) on the following coordinate grid. Time (Seconds) Elevation (feet) Let s Practice! 1. A construction company builds houses on square-shaped lots of various sizes. The CEO of the company decided to diversify her lots and now has houses built on rectangular-shaped lots that are 6 feet longer and 4 feet narrower than her square-shaped lots.

9 A. What is the function that models the size of the rectangular lots relative to the size of the square lots? = + = + b. Complete the table below and graph the function. ( ) Square Lot Size Rectangle Lot Size Try It! 2. A business owner recorded the following data for an entire year of sales. Month Sales (in thousands of dollars) Jan 22 Feb 45 Mar 54 April 63 May 70 June 71 July 70 Aug 64 Sept 54 Oct 38 Nov 24 Dec 5 a. Plot the data on the graph below. b. What type of business might be represented by this graph? Answers Vary. Sample answers: Shaved Ice Cones or Theme Park c. Would the Quadratic model be an appropriate way to model data for this business going forward?

10 Justify your answer. Yes, it follows a parabolic pattern. Month of Year Sales BEAT THE TEST! 1. Consider the following table of values. 5 4 3 1 2 4 ( ) 16 6 0 0 30 70 Which of the following is the graph corresponding to the table of values? A B C D Answer: D Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to to learn more and get started! Section 6 Topic 4 graphing Quadratics Using the Vertex and Intercepts Given a Quadratic equation in standard form, ( )= 3 4 12, use the following steps to graph ( ) on the coordinate plane on the following page.