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Graphing Quadratic Functions: Parabolas

Graphing Quadratic Functions: Parabolas. Which of the following relationships describes the pattern: 1, −1 , −3 , −5 …? A. t n = −2 n B. t n = −2 n + 3 C. t n = 2 n – 3 D. t n = 2 n. Pattern Probes. 2. What is the 62th term of the pattern: 1, 9, 17, 25, …? A. 489 B. 496

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Graphing Quadratic Functions: Parabolas

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  1. Graphing Quadratic Functions:Parabolas

  2. Which of the following relationships describes the pattern:1, −1, −3, −5 …? A. tn = −2n B. tn = −2n+ 3 C. tn = 2n – 3 D. tn = 2n Pattern Probes 2. What is the 62th term of the pattern:1, 9, 17, 25, …? A. 489 B. 496 C. 503 D. 207

  3. Which of the following relationships describes the pattern:1, −1, −3, −5 …? A. tn = −2n B. tn = −2n+ 3 C. tn = 2n – 3 D. tn = 2n Pattern Probes 2. What is the 62th term of the pattern:1, 9, 17, 25, …? A. 489 B. 496 C. 503 D. 207

  4. Quadratics Patterns Note: Another phrase for linear pattern is an arithmetic sequence. Last time we learned that: A linear pattern has a common difference on level 1. A linear pattern is given by tn = t1 + (n − 1)d A quadratic pattern has its common difference on level 2. The CD = 2a for a quadratic The quadratic pattern is defined by tn = an2 + bn + c The general form of the quadratic function is defined byf(x) = ax2 + bx + c

  5. Let’s Graph a = 1b = 0c = 0 Quadratics, as we know, do not increase by the same amount for every increase in x(or n). (How do we know this?) Therefore, their graphs are not straight lines,but curves, called parabolas. In fact, if you’veever thrown a ball, or watched a frog jumpyou’ve seen a quadratic graph! Its table of values would be: Consider the functionf(x) = x2 It’s the simplest quadratic (Why?)

  6. Let’s Graph a = 1b = 0c = 0 The pattern would be Quadratics, as we know, do not increase by the same amount for every increase in x(or n). (How do we know this?) Therefore, their graphs are not straight lines,but curves, called parabolas. In fact, if you’veever thrown a ball, or watched a frog jumpyou’ve seen a quadratic graph! Its table of values would be: Consider the functionf(x) = x2 It’s the simplest quadratic (Why?)

  7. Let’s Graph Once we have the vertex, the rest of the points follow the pattern: over 1 (from the vertex), up 1 (from the vertex) over 2(from the vertex), up 4(from the vertex) over 3(from the vertex), up 9(from the vertex) over 7(from the vertex), up 49(from the vertex)

  8. Properties of the graph y = x2 vertex:the point (0, 0) axis of symmetry:the line x = 0 domain:{x R} range:{y ≥ 0, y R}

  9. Transformations of y = x2 This simplest parabola can be: stretched vertically slid horizontally slid vertically We give these transformations short forms: VS, HT, and VT, These show up (undone) in the transformational form of the equation:

  10. Transformations Ex. Graph the function: (FYI, its general formis y = 2x2 – 4x – 1) So we can use these to get a MAPPING RULE: (x, y) →(x + 1, 2y – 3) What are the transformations that have been applied to the base graph of y = x2? Vertical Stretch VS = 2 (positive, so no reflection) Vertical Translational VT = –3 Horizontal Translational HT = 1 new x and y values oldx and y values, fromy = x2 We now take the old table of values and do what the mapping rule says.

  11. Transformations Ex. Graph the function: (FYI, its general formis y = 2x2 – 4x – 1) New x and y values for this transformed parabola “old” x and y values(from the base parabola) Apply the mapping rule: (x, y)→(x + 1, 2y – 3)

  12. Transformations Transformationson y = x2: VS = 2 VT = –3 HT = 1 Ex. Graph the function:

  13. Transformations Ex. Graph the function: Short-cut: Once we have the vertex, the rest of the points follow the pattern: over 1, up 1 ×2 over 2, up 4 ×2 over 3, up 9 ×2 . . . over #, up # ×VS (again, always from the vertex)

  14. Transformations Ex. Graph the function: vertex (1, –3)(that is (HT, VT)) axis of symmetry x = 1(that is the line x= HT) domain {x R}(unchanged) range {y≥–3,y R}(that is {y≥ VT, y R})

  15. Graphing Parabolas: Practice Graph the following. State the vertex, axis of symmetry, range, domain and the y-intercept of each. A) (or y = 2x2 – 8x + 11, in general form) B) (or y = – ½x2 – x – ½ , in general form) C) (or y = –x2 – 12x – 32, in general form)

  16. A) (or y = 2x2 – 8x + 11 in general form) domain: {x R} range: {y≥3, y R} axis of symmetry: x = 2 y-intercept (0,11) VS = 2HT = 2VT = 3 (x, y) →(x + 2, 2y + 3)

  17. B) (or y = -½x2 - x - ½ , in general form) domain: {x R} range: {y≤0, y R} axis of symmetry: x = –1y-intercept (0, – ½) VS = –½HT = –1VT = 0 (x, y) →(x – 1, – ½y)

  18. C) (or y = –x2 – 12x – 32 in general form) domain: {x R} range: {y≤ 4, y R} axis of symmetry: x = –6 y-intercept (0, –32) VS = –1 HT = –6VT = 4 (x, y) →(x – 6, – y + 4)

  19. Going backwards: Finding the Equation Find the equation of this graph: Answer:2(y + 2)=(x – 1)2 Give the general form of this equation: More on this to come….

  20. Going backwards: Finding the Equation Answers a) b) • What quadratic function has a range of {y≥6, y R}, a VS of 4 and an axis of symmetry of x = –5? • What quadratic function has a vertex of (2, – 12) and passes through the point (1, –15)?

  21. Solving for x given yusing transformational form Example:The parabola passes through the points (x, 2). What are the values of x? We can use transformational form to solve for the x-values given a certain y-value.(Note – there will (almost) always be TWO such x-values!)

  22. Solving for x given yusing transformational form Find the x-intercepts (roots) of the following quadratic functions:

  23. Solving for y given xusing transformational form Find the y-intercept of the following quadratic functions: FYI, this is much easier from standard form:

  24. Comparing Forms not coming soon… coming soon… General Form: Transformational Form: Standard Form: Factored Form:

  25. What are you good for? Transformational form is good for: • finding the vertex • (HT, VT) • finding the range • {y≤ or ≥ VT, y R} • finding the axis of symmetry • x = HT • finding the max/min value • VT • getting the mapping rule • (x, y) → (x + HT, VSy + VT) • graphing the function • getting the equation from the graph • Drawbacks: • Functions are usually written in the f(x) notation

  26. What are you good for? General form is good for: • finding the y-intercept • (0, c) • finding the x-intercepts • …coming soon…. • finding the vertex • …coming soon…. • graphing the parabola • …coming soon… • finding the roots • …coming soon… • Drawbacks: • Some work is involved to obtain this form when only graph, or only transformations are given

  27. What are you good for? • Standard form is good for: • nothing

  28. What are you good for? • Factored form is good for: • finding roots • r1 and r2 • Drawbacks: • Not all quadratic functions can be factored

  29. Going Between Forms 4 steps 2 steps • transformational → general • 2 steps: FOIL then solve for x • general → transformational • 4 steps: includes completing the square • …coming soon….

  30. Transformational to General Form FOIL irst utside nside ast

  31. Transformational to General Form Ex. Put the following into general form:

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