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Chapter 5 Prentice Hall

Chapter 5 Prentice Hall. Quadratics. 5.1/5.2 Graphing quadratic functions. Quadratic function : f(x) = ax ² + bx + c Graph is a parabola. Quadratic Term. Constant Term. Linear Term. ax ² + bx + c. y-intercept is c Where x=0 Equation of the axis of symmetry:

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Chapter 5 Prentice Hall

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  1. Chapter 5 Prentice Hall Quadratics

  2. 5.1/5.2 Graphing quadratic functions • Quadratic function: f(x) = ax² + bx + c • Graph is a parabola Quadratic Term Constant Term Linear Term

  3. ax² + bx + c • y-intercept is c Where x=0 • Equation of the axis of symmetry: • x-coordinate of the vertex:

  4. Example • Consider f(x) = x² + 9 + 8x • Find the y-intercept, equation of the axis of symmetry and the x-coordinate of the vertex. • A=1, B= 8 and C=9 • Y int = 9 • X=-b/2a • X=-4

  5. Maximum or Minimum • a>0: opens up/minimum • a<0: opens down/maximum • f(-b/2a) is value of max or min

  6. Maximum or Minimum • Max or Min can be found at the y coordinate of the vertex. • How do we find the x coordinate? • -b/2a • Then plug in x value to find y value.

  7. 5.3 Movement of Graphs • Graph f(x) = x2 and g(x) = -x2. Describe how the graphs of f(x) and g(x) and are related.

  8. Change to Parent Graph Reflections Y=-f(x) Outside the HVertical Axis Reflected over the x-axis Y=f(-x) Inside the HHorizontal Axis Reflected over the y-axis

  9. Change to Parent Graph Translations +,- OUTSIDE of Function Outside the H Vertical Movement SHIFTS UP AND DOWN +,- INSIDE of Function Inside the H Horizontal Movement SHIFTS LEFT AND RIGHT

  10. Change to Parent Graph Dilations X/÷ OUTSIDE of Function Outside the H Vertical Movement Expands/Compresses X/÷ INSIDE of Function Inside the H Horizontal Movement Expands/Compresses

  11. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = x2 + 1 • This function is of the form y= f(x) + 1. • Outside the HVertical Movement • Since 1 is added to the parent function y = x2,the graph of the parent function moves up 1 unit.

  12. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = (x - 2)2 • Inside the HHorizontal Movement • This function is of the form y = f(x - 2). • Since 2 is being subtracted from x before being evaluated by the parent function, the graph of the parent function y = x2 slides 2 units right. a.

  13. Examples - Use the parent graph y = x2 to sketch the graph of each function. • y = (x + 1)2 – 2 • This function is of the form y = f(x + 1)2 -2. The addition of 1 indicates a slide 1 unit left, and the subtraction of 2 moves the parent function y = x2 down two units. a.

  14. 5.5 Solving QE by Graphing • Roots/Zeros • Solutions of the QE • X-Intercepts • Where y=0 • Can check answers if polynomial = 0

  15. Graphing EQUATIONS

  16. Estimating Roots • Solve –x2+4x-1=0 by graphing. If exact roots can not be found, state the consecutive integers between which the roots are located • X Intercepts are where? • Between 0 and 1 and between 3 and 4

  17. 5.4 Solve QE by factoring • Zero product property – if ab = 0, then a = 0, b = 0, or both This means to factor and set each equal to zero

  18. To write an equation given roots • 1. Use (x – p)(x – q) = 0 • 2. FOIL or BOX • 3. Multiply to ensure a,b,c are integers

  19. 5.7 Completing the square • Square root property: If x² = n , then x2+10x+25=49 X2-6x+9=32

  20. CTS: put in form ax² + bx = c Find: • b = • ½ b = • (½b)² = • Add (½b)² to both sides • Re-write LHS as a perfect square • Solve • x2+bx+(b/2) 2=(x+b/2) 2

  21. When should I CTS? • You can only CTS when a = 1 • Only complete the square if b is even • x2+4x-12=0 • 3x2-2x-1=0 • x2+2x+3=0

  22. Vertex Form: y = a(x – h)² + k • Vertex: (h, k) • Axis of sym: x = h • a means: open up/down and wide/narrow

  23. Convert from quadratic form to vertex form • 1. Isolate y • 2. Complete the square on the RHS • 3. Add/subtract same # on the right • (if a ≠ 1, start by factoring a out of first two terms) • Check to see if both are same parabola in calculator

  24. To write in vertex form given vertex and a point: • 1. Use x, y, h, k to find a • 2. Plug in values of h, k, a into vertex form • Check mult. Choice by graphing and checking table

  25. 5.8 Quadratic Formula • If • Then

  26. Discriminant • If

  27. QUADRATIC EQUATIONS • b2-4ac >0 • Two distinct real roots • b2-4ac=0 • Exactly one real root (actually a double root) • b2-4ac<0 • No real roots (Two distinct imaginary roots)

  28. To graph quadratic inequalities • 1. Graph parabola • 2. Test a point inside • 3. True: shade inside False: shade outside

  29. To solve quadratic inequalities algebraically • 1. Find zeros • 2. Test points left, center, and right

  30. Number Theory • Find two real numbers whose sum is 6 and whose product is 10 or show that no such numbers exist • X=One number • 6-x=the other number • Product is 10 • X(6-X)=10 • Solve by graphing/factoring • No such numbers

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