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

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Presentation transcript:

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?  Predict how the graph of y=1/3x 2 will be similar and different to the graph of y=x 2 ?

Vocabulary:  Standard Form of a Quadratic Function = a function that can be written in the form of ax 2 +bx+c where a does not equal 0.  Quadratic Parent Function = f(x) = x 2 or y = x 2  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….

Anatomy of a Parabola

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

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

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.

Graph y=ax 2 +bx+c  Graph the function -3x 2 +6x+5.  Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex…

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

Graphing continued:  Vertex = (1,8)  Axis of symmetry = x=1

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)

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.

Try your own…  Graph f(x) = x 2 -6x+9

Recap: Conclusions  y=a 2 +bx+c  Positive a: opens up  Vertex = minimum  y=-a 2 +bx+c  Negative a: Opens down  Vertex = maximum  The larger the a, the narrower the graph.

Graphing Quadratic Inequalities  Using the last equation: Graph –  y<x 2 -6x+9  Remember: dashed line; test one point below or above the line; then, shade.

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