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D Use a table of values to graph y = x2 + 2x – 1. State the domain and range. What is the equation of the axis of symmetry for y = –x2 + 2? What are the coordinates of the vertex of the graph of y = x2 – 5x? Is the vertex a maximum or minimum? What is the maximum height of a rocket fired straight up if the height in feet is described by h = –16t2 + 64t + 1, where t is time in seconds?
Slide 2: Homework: Start Page 540
WM – 4 problems 11 – 45 odd CW – 12 problems 47 – 50 all HW – 33 problems 51 – 75 odd
Slide 3: Then/Now
New Vocabulary
Key Concept: Solutions of Quadratic Equations
Example 1: Two Roots
Example 2: Double Root
Example 3: No Real Roots
Example 4: Approximate Roots with a Table
Example 5: Real-World Example: Approximate Roots with a Calculator
Slide 4: You solved quadratic equations by factoring. (Lesson 8–3) Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing.
Slide 5: double root
Slide 7: Two Roots Solve x2 – 3x – 10 = 0 by graphing. Graph the related function f(x) = x2 – 3x – 10. The x-intercepts of the parabola appear to be –2 and 5. So the solutions are –2 and 5.
Slide 8: Two Roots Check Check each solution in the original equation. x2 – 3x – 10 = 0 Original equation x2 – 3x – 10 = 0 0 = 0 Simplify. 0 = 0
Slide 9: Double Root Solve x2 + 8x = –16 by graphing. Step 1 First, rewrite the equation so one side is equal to zero. x2 + 8x = –16 Original equation
x2 + 8x + 16 = –16 + 16 Add 16 to each side.
x2 + 8x + 16 = 0 Simplify.
Slide 10: Double Root Step 2 Graph the related function f(x) = x2 + 8x + 16.
Slide 11: Double Root Step 3 Locate the x-intercepts of the graph. Notice that the vertex of the parabola is the only x-intercept. Therefore, there is only one solution, –4.
Slide 12: No Real Roots Solve x2 + 2x + 3 = 0 by graphing. Graph the related function f(x) = x2 + 2x + 3. The graph has no x-intercept. Thus, there are no real number solutions for the equation.
Slide 13: Approximate Roots with a Table Solve x2 – 4x + 2 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. Graph the related function f(x) = x2 – 4x + 2.
Slide 14: Approximate Roots with a Table The x-intercepts are located between 0 and 1 and between 3 and 4.
Make a table using an increment of 0.1 for the x-values located between 0 and 1 and between 3 and 4.
Look for a change in the signs of the function values. The function value that is closest to zero is the best approximation for a zero of the function.
Slide 15: Approximate Roots with a Calculator MODEL ROCKETS Consuela built a model rocket for her science project. The equation h = –15.6t2 + 250t models the flight of the rocket, launched from ground level at a velocity of 250 feet per second, where h is the height of the rocket in feet after t seconds. Approximately how long was Consuela’s rocket in the air? You need to find the roots of the equation –15.6t2 + 250t = 0. Use a graphing calculator tograph the related function h = –15.6t2 + 250t.
Slide 16: Approximate Roots with a Calculator The x-intercepts of the graph are approximately 0 and 16 seconds. Answer: The rocket is in the air approximately 16 seconds.
Slide 17: Your Turn
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D Solve x2 – 2x – 8 = 0 by graphing.
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D Solve x2 + 2x = –1 by graphing.
Slide 20: Solve x2 + 4x + 5 = 0 by graphing.
Slide 21: Solve x2 – 5x + 1 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
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D GOLF Martin hits a golf ball with an upward velocity of 120 feet per second. The function h = –16t2 + 120t models the flight of the golf ball hit at ground level, where h is the height of the ball in feet after t seconds. How long was the golf ball in the air?
Slide 23: Homework: Start Page 540
WM – 4 problems 11 – 45 odd CW – 12 problems 47 – 50 all HW – 33 problems 51 – 75 odd