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Finding the square root button on your online quiz/test calculator for all quizzes/tests that include problems from Ch. 10 and 11: Remember: The calculator button is at the TOP LEFT of your screen when you are in the lockdown browser, not at the right side of the problem like it is when you’re doing online homework. This version of the calculator will be also be available when you take Test 4 and the Final Exam. Please open Quiz 11.2 You will have access to the online calculator on your laptop during this quiz. No other calculator may be used. You may use the pink formula sheet on this quiz – please don’t write on this sheet, and remember to hand it back in with your quiz answer sheet. Remember to turn in your answer sheet to the TA when the quiz time is up. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note-taking materials. Coming up: • Tomorrow: • HW 11.5 is due at start of class • In-class quiz on HW 11.5 • Lecture: Review for Test 3 • Monday, 12/1: • Take Test 4 (60 points, on last 2 weeks’ material) • Practice Test 4 is due at start of class on Monday. • Next Tuesday, Wednesday and Thursday: • Review lectures on Units 1-3 (Tests 1-3) • Review HW on each unit, worth double points (8 pts each) • Make sure you know when the final exam is scheduled for this section, and mark it on your calendar. Section 11.5 Graphing Quadratic Equations We first examined the graph of f(x) = x2 back in Chapter 3, and you graphed some variations of the quadratic function f(x) = ax2 + bx + c on the graphing worksheets for HW 8.2 A & B. (You may want to refer to your graded worksheets as you do the homework for this section.) Reminders on graphing quadratic functions of the form f(x) = ax2 + c . Recall from the HW 8.2 worksheets: • If a > 0, the parabola opens upward. • If a < 0, the parabola opens downward. • The point (0, C) is the y-intercept of the graph. Examples y f(x) = x2 f(x) = -x2 g(x) = x2 + 3 x h(x) = x2 – 3 Two new terms we’ll be introducing in this section: • The highest point or lowest point on the parabola is called the vertex. • The axis of symmetry is the line that runs through the vertex and through the middle of the parabola. Examples y f(x) = x2 Vertex = (0,0) f(x) = -x2 Vertex = (0,0) g(x) = x2 + 3 Vertex = (0,3) x h(x) = x2 – 3 Vertex = (0,-3) The axis of symmetry for all four of these graphs is the vertical line x = 0 Graphing the parabola f(x) = (x – h)2 • If h is positive, the graph of f(x) = (x – h)2 is the graph of y = x2 shifted to the right h units. • If h is negative, the graph of f(x) = (x – h)2 is the graph of y = x2 shifted to the left |h| units. • The vertex is (h, 0) . • The axis of symmetry is the vertical line x = h. Example (cont) f(x) = x2 5 y 4 g(x) = (x – 3)2 3 2 Vertex: (3, 0) 1 x Axis: x = 3 -5 -4 -3 -2 -1 0 -1 -2 h(x) = (x + 3)2 -3 -4 Vertex: (3, 0) Axis: x = 3 -5 1 2 3 4 5 Graphing the parabola f(x) = (x – h)2 + k • The parabola has the same shape as y = x2. • The vertex is the point (h, k). • The axis of symmetry is the vertical line x = h. Example (cont) f(x) = x2 y g(x) = (x – 2)2 + 4 Vertex: (2, 4) Axis: x = 2 x Graphing the parabola f(x) = ax2 • If a is positive, the parabola opens upward • If a is negative, the parabola opens downward. • If |a| > 1, the graph of the parabola is narrower than the graph of y = x2. • If |a| < 1, the graph of the parabola is wider than the graph of y = x2. Example (cont) y f(x) = x2 g(x) = 3x2 h(x) = x (1/3)x2 The vertex of all three of these graphs is the point (0,0). The axis of symmetry for all three of these graphs is the vertical line x = 0 Example Graph g(x) = –4(x + 2)2 – 1. Find the vertex and axis of symmetry. • Rewrite the function: g(x) = –4(x – (–2))2 – 1. • The graphs opens down and is narrower than f(x) = x2 . • The graph is the graph of f(x) = x2 shifted two units to the left and one unit down. • Vertex: (–2, –1) • Axis: x = –2 5 y 4 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 x 3 4 5 Using the online graphing for tool. this assignment: (7, 7) First, select the tool parabola (10,16) Next, find a second point on the graph by picking a number to plug in for x and calculating the y coordinate. Pick an x that is not part of the vertex point, and choose it so that the x and y values can be plotted on the scale of the graph. (More than one choice will work.) For example, if we choose x = 10, then y = (10 - 7)2 + 7 = 32 + 7 = 16. Then (10,16) would be a second point we can plot on the parabola. Next, select the line tool. Use the line tool to graph the axis of symmetry. Pay attention to this next part!! After you graph the axis of symmetry line, you have to change it to a dotted line in order for the software to know that you’ve got the correct answer. (7, 7) • Once you have the parabola AND the axis of symmetry graphed, and have changed the axis to a dotted line, click “save”. • Then you’ll be asked to type in the vertex as an ordered pair. • That would be (7,7) in the previous example. • Finally, you’ll be asked to give the equation of the axis of symmetry. • That would be x = 7 in the previous example. • You can now open your laptops and try using this tool on HW 11.5 Make sure you can successfully use the tool (i.e. “check answer” and get it right) before you leave class today. REMINDER: The assignment on today’s material (HW 11.5) is due at the start of the next class session. If time remains, please open your laptops and work on the homework assignment until the end of the class period. Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m.