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Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations

Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up : Factor 1. 49p 2 – 100 2. 6d 4 + 4d 3 – 6d 2 – 4d Solve for x: 3. 2x 2 + 13x + 6. Quadratic Function (y = ax 2 + bx + c). y = x 2 – x – 2.

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Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations

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  1. Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up: Factor1. 49p2 – 100 2. 6d4 + 4d3 – 6d2 – 4d Solve for x: 3. 2x2 + 13x + 6

  2. Quadratic Function(y = ax2 + bx + c) y = x2 – x – 2 y = 3x2 y = x2 + 9

  3. Vocabulary: • Quadratic Parent Function • Parabola = the graph of a quadratic function is a U-shaped curved. • Axis of Symmetry – divide the graph into two halves The line of symmetry ALWAYS passes through the vertex. Continue

  4. 4. Vertex • Minimum – lowest point of the parabola • Maximum – the highest point of the parabola. y Vertex Maximum x Vertex Minimum

  5. y = ax2 + bx + c y = x2 • a = 1, b = 0, c = 0 • Minimum point (0,0) • Axis of symmetry x=0 y=x2

  6. Finding the Line of Symmetry When a quadratic function is in standard form For example… Find the line of symmetry of y = 3x 2 – 18x + 7 y = ax2 + bx + c, The equation of the line of symmetry is Using the formula… Thus, the line of symmetry is x = 3. Finding the Vertex What is the vertex? How do I find my vertex?

  7. Another example: y = –2x 2 + 8x –3 STEP 1: Find the line of symmetry We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. STEP 2: Plug the x – value into the original equation to find the y value. y = –2(2)2 + 8(2) –3 To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5)

  8. A Quadratic Function in Standard Form Let's Graph ONE! Try … y = 2x 2 – 4x – 1 STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find the y-intercept. STEP 4: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

  9. y x A Quadratic Function in Standard Form Step 5:Lets Graph it! y = 2x2 – 4x – 1

  10. What happen if we change the value of a and c ? y=3x2 y=4x2+3 y=-4x2-2 y=-3x2

  11. When a is positive, When a is negative, When c is positive When c is negative the graph concaves downward. the graph concaves upward. the graph moves up. the graph moves down. Conclusion to Quadratic Function(y = ax2+bx+c)

  12. Solving Quadratic Equations Method #1: • Quadratic Formula x2 – 2x – 8 = 0 Hint: Quadratic equation, must equal 0

  13. Example #2 x2 – 4x = 21

  14. Method #2: Factoring x2 - 2x = 0 • Factor in order to solve the equation. y=x2-2x • Hints: • Remember to ask yourself does the function have a GCF. • Find the x intercept. Answer: Two solutions, x=0 and x=2.

  15. Group Work: Page 544 – 545 Group 1: #7, #20 Group 2: #8, #21 Group 3: #9, #22 Group 4: #10, #23 Group 5: #11, #24 Group 6: #12, #25 Group 7: #13, #28 Group 8: #14, #30 Independent Work: Page 538 (8, 10) Page 546 (43) Page 549 (a, b, c)

  16. More Practice Solve the following equations by factoring: • x2 = 25 • x2 – 8 = - 7x • x2 – 12x = -36 • x2 = 7x 5. m2 – 3m = 10 6. 2r – 8 = - r2 7. 4x2 = 49 8. x2 = 9x – 20

  17. y = -x2 + 2x + 8

  18. Find the Solutions y=x2-4 y=x2+2x-15 y=-x2+5 y=-x2-1

  19. Find the solutions y=-x2+4x-1 y=x2+2x+1

  20. HLA#5: Page 538 (7, 8) Page 544 (2, 3, 4)

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