Graphing Quadratic Equations Objective: SWBAT graph quadratic equations
Parabolas Examples The path of a jump shot as the ball travels toward the basket is a parabola.
Key terms Parabola – a curve that can be modeled with a quadratic function. Quadratic function – a function that can be written in the form
Key terms - continued Vertex – the point where a parabola crosses its line of symmetry. Maximum – the vertex of a parabola that opens downward. The y-coordinate of the vertex is the maximum value of the function. Minimum – the vertex of a parabola that opens upward. The y-coordinate of the vertex is the minimum value of the function. y-intercept – the y-coordinate of the point where a graph crosses the y-axis. x-intercept – the x-coordinate of the point where a graph crosses the x-axis. Line of symmetry
Lucky for you… …graphing parabolas won’t get that difficult. For right now, we are sticking with parabola shifts. given: y = (x – h)2 + k h shifts the parabola a units to the right -h shifts the parabolas a units to the left k shifts the graph up b units -k shifts the graph down b units Integrated 2 4-1 Graphing Quadratic Functions
Example 1
Direction and Min/Max The graph of the quadratic function , is a parabola. If a is positive the graph opens up the vertex is a minimum If a is negative the graph opens down the vertex is a maximum Note that the parabola does not have a constant slope. In fact, as x increases by 1 , starting with x = 0 , y increases by 1, 3, 5, 7,… . As x decreases by 1 , starting with x = 0 , y again increases by 1, 3, 5, 7,… .
Example 1
Example 1 - Answers
Graphing y = a(x - h)2 + k In addition to shifting the parabola up, down, left, and right, we can stretch or shrink the parabola vertically by a constant. We can make a data table for the graph of y = 2x 2 : Data Table for y = 2x 2 Here, the y increases from the vertex by 2, 6, 10, 14,… ; that is, by 2(1), 2(3), 2(5), 2(7),… .
In general, in the graph of y = a(x - h) + k , as x increases or decreases by units of 1 starting from the vertex, y increases by 1a, 3a, 5a, 7a,… .
Vertical Shrink or Stretch? Reflection over x-axis? Summarize g(x) = ax2 Value of a Vertical Shrink or Stretch? Reflection over x-axis? a > 1 0 < a < 1 -1 < a < 0 a = 1 a < -1
Example 2
Example 2 - Answers
Putting it all together… y = a(x-h)2 + k