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10.1 & 10.2: Exploring Quadratic Graphs and Functions

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10.1 & 10.2: Exploring Quadratic Graphs and Functions

Objective: To graph quadratic functions

Linear Equations?

Quadratic Equations?

Exponential Equations?

- 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?

- 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….

- The larger the a…..?
- The smaller the a….?

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

- 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.

- Graph the function -3x2+6x+5.
- Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex…

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

- Vertex = (1,8)
- Axis of symmetry = x=1

- 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)

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

- Graph f(x) = x2-6x+9

- 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.

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

- Practice 10-1 #1-18 and 10-2#4-15, 22-24