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Learn about solutions of quadratic equations, graphing methods, axis of symmetry, and real-world examples. Practice finding roots and using a table to solve. Understand mathematical relationships and key features of graphs. Enhance problem-solving skills in under five minutes!

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  1. Five-Minute Check (over Lesson 3–1) Mathematical Practices Then/Now New Vocabulary Example 1: Two Real Solutions Key Concept: Solutions of a Quadratic Equation Example 2: One Real Solution Example 3: No Real Solution Example 4: Estimate Roots Example 5: Solve by Using a Table Example 6: Real-World Example: Solve by Using a Calculator Lesson Menu

  2. Does the function f(x) = 3x2 + 6x have a maximum or a minimum value? A. maximum B. minimum 5-Minute Check 1

  3. Find the y-intercept of f(x) = 3x2 + 6x. A. –1 B. 0 C. 1 D. 2 5-Minute Check 2

  4. Find the equation of the axis of symmetry for f(x) = 3x2 + 6x. A.x = y + 1 B.x = 2 C.x = 0 D.x = –1 5-Minute Check 3

  5. Find the x-coordinate of the vertex of the graph of the functionf(x) = 3x2 + 6x. A. 1 B. 0 C. –1 D. –2 5-Minute Check 4

  6. A. ans B. C. ans D. ans Graph f(x) = 3x2 + 6x. 5-Minute Check 5

  7. A.y = 2x2 – 4x + 3 B.y = –x2 + 2x – 1 C.y = x2 + x + 1 D.y = 3x2 – 6x 1 __ 2 Which parabola has its vertex at (1, 0)? 5-Minute Check 6

  8. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. MP

  9. You solved systems of equations by graphing. • Solve quadratic equations by graphing. • Estimate solutions of quadratic equations by graphing. Then/Now

  10. quadratic equation • standard form • root • zero Vocabulary

  11. Two Real Solutions Solve x2 + 6x + 8 = 0 by graphing. Graph the related quadratic function f(x) = x2 + 6x + 8. The equation of the axis of symmetry is x = –3. Make a table using x-values around –3. Then graph each point. Example 1

  12. ? ? (–4)2 + 6(–4) + 8 = 0 (–2)2 + 6(–2) + 8 = 0 Two Real Solutions We can see that the zeros of the function are –4 and –2. Answer: The solutions of the equation are –4 and –2. Check Check the solutions by substituting each solution into the original equation to see if it is satisfied. x2 + 6x + 8 = 0 x2 + 6x + 8 = 0 0 = 0 0 = 0   Example 1

  13. A.B. C.D. –3, 1 –1, 3 –3, 1 –1, 3 Solve x2 + 2x – 3 = 0 by graphing. Example 1

  14. Concept

  15. x2 – 4x = –4 x2 – 4x + 4 = 0 Add 4 to each side. One Real Solution Solve x2 – 4x = –4 by graphing. Write the equation in ax2 + bx + c = 0 form. Graph the related quadratic function f(x) = x2 – 4x + 4. Example 2

  16. One Real Solution Notice that the graph has only one x-intercept, 2. Answer: The only solution is 2. Example 2

  17. A.B. C. D. 3 3 –3 –3 Solve x2 – 6x = –9 by graphing. Example 2

  18. No Real Solution NUMBER THEORYUse a quadratic equation to find two numbers with a sum of 4 and a product of 5. Understand Let x = one of the numbers. Then 4 – x = the other number. Plan x(4 – x) = 5 The product is 5. 4x – x2 = 5 Distributive Property x2 – 4x + 5 = 0 Add x2 and subtract 4x from each side. SolveGraph the related function. Example 3

  19. No Real Solution The graph has no x-intercepts. This means that the original equation has no real solution. Answer: It is not possible for two real numbers to have a sum of 4 and a product of 5. CheckTry finding the product of several numbers whose sum is 4. Example 3

  20. NUMBER THEORYUse a quadratic equation to find two numbers with a sum of 7 and a product of 14. A. 7, 2 B. –7, –2 C. 5, 2 D. no such numbers exist Example 3

  21. Estimate Roots Solve –x2 + 4x – 1 = 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Make a table of values and graph the related function. Example 4

  22. Estimate Roots The x-intercepts of the graph are between 0 and 1 and between 3 and 4. Answer: One solution is between 0 and 1 and the other is between 3 and 4. Example 4

  23. Solve x2 – 4x + 2 = 0 by graphing. What are the consecutive integers between which the roots are located? A. 0 and 1, 3 and 4 B. 0 and 1 C. 3 and 4 D. –1 and 0, 2 and 3 Example 4

  24. Solve by Using a Table Solve x2 + 5x – 7 = 0. Enter y1 = x2 + 5x – 7 in your graphing calculator. Use the TABLE window to find where the sign of Y1 changes. Change ΔTbl to 0.1 and look again for the sign change. Replace the process with 0.01 and 0.001 to get a more accurate location of the zero. Example 5

  25. Solve by Using a Table Answer: One solution is approximately 1.140. Example 5

  26. Locate the second zero in the function x2 + 5x – 7 = 0 from Example 5. A. –1.140 B. –3.140 C. –5.140 D. –6.140 Example 5

  27. Solve by Using a Calculator ROYAL GORGE BRIDGE The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck is 1053 feet above the river. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) = –16t2 + h0, where t is the time in seconds and h0 is the initial height above the water in feet. We need to find t when h0 = 1056 and h(t) = 0. Example 6

  28. Use the ZERO feature, [CALC], to find the positive zero of the function, since time cannot be negative. Use the arrow keys to locate a left bound for the zero and press . 2nd ENTER ENTER Then locate a right bound and press twice. Solve by Using a Calculator Solve 0 = –16t2 + 1056. Graph the related function y = –16t2 + 1056 using a graphing calculator. Adjust your window so that the x-intercepts are visible. Example 6

  29. Solve by Using a Calculator Answer: The positive zero of the function is approximately 8. It should take about 8 seconds for the marble to reach the surface of the water. Example 6

  30. HOOVER DAMOne of the largest dams in the United States is the Hoover Dam on the Colorado River, which was built during the Great Depression. The dam is 726.4 feet tall. Suppose a marble is dropped over the railing from a height of 6 feet above the top of the dam. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) = –16t2 + h0, where t is the time in seconds and h0 is the initial height above the water in feet. A. about 6 seconds B. about 7 seconds C. about 8 seconds D. about 10 seconds Example 6

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