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Ch 3 review

Ch 3 review. Quarter test 1 And Ch 3 TEST. Where a , b , and c are real numbers and a 0. Graphs of Quadratic Functions. Standard Form. Domain: all real numbers Range: depends on the minimum and maximum The graph is a parabola. if. positive. x = h. opens: axis of symmetry:

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Ch 3 review

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  1. Ch 3 review Quarter test 1 And Ch 3 TEST

  2. Where a, b, and c are real numbers and a 0 Graphs of Quadratic Functions Standard Form • Domain: all real numbers • Range: depends on the minimum and maximum • The graph is a parabola.

  3. if positive x = h opens: axis of symmetry: vertex: k is the range: up x = h (h, k) minimum V(h, k) / minimum The graph of x2 is shifted “h” units horizontally and “k” units vertically.

  4. if negative V(h, k) / maximum opens: axis of symmetry: vertex: k is the range: down x = h (h, k) maximum x = h The graph of x2 is shifted “h” units horizontally, “k” units vertically, and reflected over x-axis.

  5. Vertex form: Standard form: can “see” the transformations… The vertex form is easier to graph…to change from standard form to vertex form, either complete the square(YUCK!) or memorize this formula: and h k = f(h) Therefore, the vertex is at and the axis of symmetry is .

  6. Example: Let . Find the vertex, axis of symmetry, the minimum or maximum value, and the intercepts. Use these to graph f(x). State the domain and range and give the intervals of increase and decrease. Then write the equation in vertex form and list the transformations that were made to the parent function, f(x) = x2. a = 3 b = 6 c = 1 1st identify a, b, and c: So, the vertex is (-1, -2) and the axis of symmetry is x = -1. Since a > 0, then the graph opens up and has a minimum value at -2. Next find h and k: The intercepts are at (0, 1), (-0.184, 0), and (-1.816, 0). To find y-intercepts evaluate f(0): To find x-intercepts (roots/zeros) use the quadratic formula: -0.184 and -1.816

  7. To graph, plot the vertex, intercepts, utilize the axis of symmetry. y-int: (0,1) V(-1, -2) axis of sym: x = -1 So, to be symmetrical, another point will be at (-2, 1). Check using your graphing calculator! Domain: all real numbers Decreasing: Increasing: Range: Vertex form: f(x) = a(x - h)2 + k It is the graph of x2 shifted left 1, vertically stretched by 3 and shifted down 2. a = 3 h = -1 k = -2

  8. Example: Find the standard form equation of the quadratic function whose vertex is (1, -5) and whose y-intercept is -3. (0, -3) k h Fill in the information that was given: Vertex form: f(x) = a(x - h)2 + k Solve for a… Write the equation in vertex form then simplify to standard form:

  9. Power Functions The polynomial that the graph resembles (the end behavior model)… EX: The power function of the polynomial is…

  10. Properties of Power Functions with Even Degrees 1. f is an even function a. The graph is symmetric with respect to the y-axis b. f(-x) = f(x) 2. Domain: all real numbers 3. The graph always contains the points (0, 0), (1, 1) and (-1, 1) 4. As the exponent increases in magnitude, the graph becomes more vertical when x < -1 or x > 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis. * The graph always contains the points (0, 0), (1, 1) and (-1, 1) *Points used to make transformations

  11. EX: Graph y = x4, y = x8 and y = x12all on the same screen. Let and be your viewing window. What do you notice?

  12. Properties of Power Functions with Odd Degrees 1. f is an odd function a. The graph is symmetric with respect to the origin b. f(-x) = -f(x) 2. Domain: all real numbers Range: all real numbers 3. The graph always contains the points (0, 0), (1, 1) and (-1, -1) 4. As the exponent increases in magnitude, the graph becomes more vertical when x < -1 or x > 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis. * The graph always contains the points (0, 0), (1, 1) and (-1, -1) *Points used to make transformations

  13. EX: Graph y = x3, y = x7 and y = x11all on the same screen. Let and be your viewing window. What do you notice?

  14. Graphs of polynomial functions are smooth (no sharp corners or cusps) and continuous (no gaps or holes…it can be drawn without lifting your pencil)… Is a polynomial graph Is not a polynomial graph

  15. We can apply what we learned about transformations in Chapter 2 and what we just learned about power functions to graph polynomials…

  16. EXAMPLE: Graph f(x) = 1 – (x – 2)4using transformations. Shift right 2 units Step 1: y = x4 Step 2: y = (x – 2)4 Start with (0, 0), (1, 1) & (-1, 1) Shift up 1 unit Step 4: y = 1 – (x – 2)4 Step 3: y = - (x – 2)4 Reflect over x-axis

  17. EXAMPLE: Graph f(x) = 2(x + 1)5using transformations. Check your work with your graphing calculator. x5…(0, 0), (1, 1) & (-1, -1) (x + 1)5…shift left 1 unit 2(x + 1)5…vertical stretch by factor 2 multiply the y-values by 2

  18. Zeros and the Equation of a Polynomial Function If f is a polynomial function and r is a real number for which f(r)= 0, then r is called a real zero of f, or a root of f. If r is a real zero/root of f then: a. r is an x-intercept of the graph of f, and b. (x – r) is a factor of f In other words…if you know a zero/root, then you know a factor…if you know a factor, then you know a zero/root EX: If (x – 4) is a factor, then 4 is a zero/root… If -3 is a zero/root, then (x + 3) is a factor…

  19. EXAMPLE: Find a polynomial of degree 3 with zeros -4, 1, and 3. (Let a = 1) If x = -4, then the factor that solves to that is… If x = 1, then the factor that solves to that is… If x = 3, then the factor that solves to that is…

  20. Rational Function A function of the form , where p and q are polynomial functions and q is not the zero polynomial. The domain is the set of all real numbers EXCEPT those for which the denominator q is zero. * EX: * Enter in calculator as (x + 1)/(x – 2)...MUST put parentheses!

  21. Domain and Vertical Asymptotes To find the domain of a rational function, find the zeros of the denominator…this is where the denominator would be zero…this is where x cannot exist. The vertical asymptote(s) of a rational function are where x cannot exist…it is the virtual boundary line on the graph. Vertical asymptotes are defined by the equation ‘x =‘

  22. How the graph reacts on either side of a vertical asymptote: Goes in opposite directions as it approaches the asymptote: Goes in the same direction as it approaches the asymptote: THE GRAPH WILL NEVER CROSS THE VERTICAL ASYMPTOTE!!!

  23. The domain is and the VA is The domain is and the VA is The domain is and the VA is EXAMPLE: Find the domain and vertical asymptotes of the rational functions. a. The graph will not exist where the denominator equals zero! x + 5 = 0 x = -5 When x = -5, the graph will not exist!

  24. EXAMPLE: Find the domain and vertical asymptotes of the rational functions. b. c. x2 + 1 = 0 x2 = -1 x = not real x2 – 4 = 0 (x + 2)(x – 2) = 0 x = -2 x = 2 Domain: VA: Domain: all real #’s VA: none

  25. Intercepts on the x and y axes To find the y-intercepts of a rational function, that is where x = 0, evaluate f(0). To find the x-intercepts of a rational function, first make sure the function is in lowest terms, that is the numerator and denominator have no common factors. Then, find the zeros of the numerator. factor top & bottom first!! The zeros of the numerator are the x-intercepts (zeros) of the rational function.

  26. The y-intercept is at and the x-intercepts are at and EXAMPLE: Find the x and y intercepts of the rational functions. a. No common factors…in lowest terms. y-intercept: x-intercept:

  27. The y-intercept is at and there are no x-intercepts EXAMPLE: Find the x and y intercepts of the rational functions. b. No common factors…in lowest terms. y-intercept: x-intercept: none…the numerator has no x in it to solve for!

  28. EXAMPLE: Find the x and y intercepts of the rational functions. Cannot be factored…in lowest terms. c. y-intercept: x-intercept: The y-intercept is at 0 and the x-intercept is at 0.

  29. Domain: VA: x = 0 Graphing Rational Functions Using Transformations a. Analyze the graph of 1st find the domain and any vertical asymptotes: x2 = 0 x = 0 Next, find the x & y-intercepts: x-intercepts: none y-intercepts: none

  30. Graphing Rational Functions Using Transformations a. Analyze the graph of Is it even? It is an even function, so it is symmetric to the y-axis. To graph without using a calculator, identify a few points on the graph by plugging in x-values: f(1) = 1 f(-1) = 1 f(2) = ¼ f(-2) = ¼

  31. It is the graph of shifted right 2 and up 1. Graphing Rational Functions Using Transformations b. Use transformations to graph Shift the VA and points right 2 and up 1... Check the domain and any vertical asymptotes: (x – 2)2 = 0 x – 2 =0 x = 2 Domain: VA: x = 2 Next, check the y-intercept: y-intercepts: 1.25

  32. VA: x = 2 Graphing Rational Functions Using Transformations c. Analyze the graph of and use it to graph It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1. Find the domain and any vertical asymptotes: Domain: x - 2 = 0 x = 2 Next, find the x & y-intercepts: x-intercepts: 3 x – 3 = 0 x = 3

  33. VA: x = 2 Graphing Rational Functions Using Transformations c. Analyze the graph of and use it to graph It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1. Find the domain and any vertical asymptotes: Domain: x - 2 = 0 x = 2 Next, find the x & y-intercepts: x-intercepts: 3 y-intercepts: 1.5

  34. Properties of Rational Functions

  35. Holes (Points of Discontinuity) x-values for a rational function that cannot exist, BUT are not asymptotes. These occur whenever the numerator and denominator have a common factor. Must factor both top and bottom first!!

  36. EXAMPLE: Find the domain and vertical asymptote(s). a hole occurs at x + 1 = 0 Domain: A hole occurs at x = -1 VA: x = -3

  37. EXAMPLE: Find the domain and vertical asymptote(s). a hole occurs at x - 2 = 0 Domain: Domain: A hole occurs at x = 2 A hole occurs at x = -1 VA: x = -3 VA: x = -2

  38. Horizontal Asymptotes describe a certain behavior of the graph as or as , that is its end behavior. How the graph behaves on the far ends of the x-axis. Always written as y = The graph of a function may intersect a horizontal asymptote. The graph of a function will never intersect a vertical asymptote.

  39. Horizontal Asymptote: Three Types of Rational Functions Balanced…the degree of the numerator and denominator are equal H.A. The horizontal asymptote is where y = 2.

  40. Horizontal Asymptote: Three Types of Rational Functions Bottom Heavy…the degree of the denominator is larger than the degree of the numerator. The horizontal asymptote is where y = 0.

  41. Three Types of Rational Functions Top Heavy…the degree of the numerator is larger than the degree of the denominator Has NO HORIZONTAL ASYMPTOTE has an oblique asymptote instead… There is no horizontal asymptote.

  42. an asymptote that is neither vertical nor horizontal, but also describes the end behavior of a graph. Has the equation “y =“ and has an x in it. It is found by dividing the polynomial: top bottom (quotient only) Oblique (Slant) Asymptote Top Heavy rational functions have oblique or slant asymptotes instead of a horizontal asymptote.

  43. EXAMPLE: Find the oblique asymptote of divide the polynomials using long division… ignore remainder The oblique asymptote is y = x2 + 1. Note: The textbook considers only linear equations oblique asymptotes…

  44. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes. a. Balanced 1st find the horizontal asymptote D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ Now find domain and vertical asymptotes x = 2, x = -2 x + 2 = 0 x – 2 = 0 x = -2 x = 2 y = 2

  45. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes. a. Find the x-intercepts: D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ 2x + 1 = 0 x + 1 = 0 x = -1/2 x = -1 -1/2, -1 Find the y-intercepts: x = 2, x = -2 y = 2

  46. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes. b. Bottom-heavy Find domain and vertical asymptotes x + 1 = 0 x – 1 = 0 x = -1 x = 1 D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ Find the x-intercepts: 3 x - 3 = 0 x = 3 3 x = 1, x = -1 Find the y-intercepts: y = 0

  47. EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes. D: ___________________ x-int: _________________ y-int: _________________ VA: _________________ HA: _________________ c. Top-heavy Find the oblique asymptote: None y = 2x - 6 oblique asymptote: __________ y = 2x - 6

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