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10.1 & 10.2: Exploring Quadratic Graphs and Functions

10.1 & 10.2: Exploring Quadratic Graphs and Functions. Objective: To graph quadratic functions. Review:. Linear Equations? Quadratic Equations? Exponential Equations?. Activity:. Graph: y=x 2 and y=3x 2 on the same coordinate plane. How are they the same? How are they different?

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10.1 & 10.2: Exploring Quadratic Graphs and Functions

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  1. 10.1 & 10.2: Exploring Quadratic Graphs and Functions Objective: To graph quadratic functions

  2. Review: Linear Equations? Quadratic Equations? Exponential Equations?

  3. Activity: • Graph: y=x2 and y=3x2 on the same coordinate plane. • How are they the same? How are they different? • Predict how the graph of y=1/3x2 will be similar and different to the graph of y=x2?

  4. Vocabulary: • Standard Form of a Quadratic Function = a function that can be written in the form of ax2+bx+c where a does not equal 0. • Quadratic Parent Function = f(x) = x2 or y = x2 • Parabola: U-shaped curve = the graph of a quadratic function • Axis of Symmetry = The line that divides the parabola into 2 matching halves. • Vertex = The highest or lowest point of a parabola • Minimum/Maximum….

  5. Anatomy of a Parabola

  6. Compare the widths of parabolas.. • The larger the a…..? • The smaller the a….?

  7. Graph: • Graph y=2x2 and y=2x2+3 and y=2x2-4 on a piece of graph paper. • What conclusion can you make about ‘c’?

  8. More Rules: • Review: X = is a vertical or horizontal line? • Vertex of a parabola is the point…? • Axis of symmetry divides the parabola in half at what point? • Axis of symmetry of a quadratic function: x = -b/2a which is also the x-coordinate of the vertex.

  9. Graph y=ax2+bx+c • Graph the function -3x2+6x+5. • Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex…

  10. Graphing continued… • Axis of symmetry = -6/2(-3) = 1 • Plug in 1 for x and solve for the y-coordinate of the vertex.

  11. Graphing continued: • Vertex = (1,8) • Axis of symmetry = x=1

  12. Graphing Continued… • Find 2 other points on the graph. • 1. Use the y-intercept where x = 0. • When x=0, y=5 • 2. Try another point on the same side of the vertex as the y-intercept… Let x = -1 • When x=-1, y=-4 so another point is (-1, -4)

  13. Graphing Continued… • Step 3: Reflect the points (0,5) and (-1, -4) across the axis of symmetry to get 2 more points… Draw the parabola.

  14. Try your own… • Graph f(x) = x2-6x+9

  15. Recap: Conclusions • y=a2+bx+c • Positive a: opens up • Vertex = minimum • y=-a2+bx+c • Negative a: Opens down • Vertex = maximum • The larger the a, the narrower the graph.

  16. Graphing Quadratic Inequalities • Using the last equation: Graph – • y<x2-6x+9 • Remember: dashed line; test one point below or above the line; then, shade.

  17. Homework: • Practice 10-1 #1-18 and 10-2#4-15, 22-24

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