Henley Task teaches horizontal transformations Protein Bar Toss Part 1 teaches factoring if a ≠ 1 Section 3.4 for a = 1 Section 3.5 for a ≠ 1 Protein Bar Toss Part 2 teaches changing from standard and vertex forms

Math II, Sections 3.1 – 2.4 Standard: MM2A3c Students will Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change.

Quadratic Function  A quadratic function is a function that can be written in the standard form: y = ax 2 + bx + c, where a ≠ 0  The graph of a quadratic equation is a parabola. The lowest or highest point on a parabola is the vertex. The axis of symmetry divides the parabola into mirror images and passes through the vertex.

 A quadratic function is a function  What is a function?  The graph of a quadratic equation is a parabola  What is a parabola, what does it look like?  The lowest or highest point on a parabola is the vertex  What does it mean when we say the lowest or highest point?  The axis of symmetry divides the parabola into mirror images  What are mirror images

Axis of Symmetry of Quadratic  The quadratic function is a symmetrical function around a vertical axis of symmetry. That means, if we draw a vertical line through the function, the distance from the axis of symmetry to the function in both directions is the same.

Axis of Symmetry of Quadratic  Graph (using an “H” table), calculate the zeros and compare axis of symmetry of the following functions: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 h(x) = x 2 + 6x - 7 (later) Show some Geosketch examples  Explain your findings and make a statement about what c does in the equation of ax 2 + bx + c = 0

Calculating the Axis of Symmetry of Quadratics  The c in the standard form of the quadratic equation ax 2 + bx + c = 0, simply moves the graph vertically. It does not change the axis of symmetry.  Since c can be changed without changing the axis of symmetry, let us choose c to equal zero and find the zeros of the resulting equation and the axis of symmetry.

 We now have: ax 2 + bx = 0  Factoring out GCF gives:  Solving gives  We also know the axis of symmetry is the vertical line in the center of the zeros, so the axis of symmetry is at the mean (average) of the two zeros.  The axis of symmetry is located at: Axis of Symmetry of Quadratic x(ax + b) = 0 x = 0 or x = -b/a x = (–b/a + 0)/2 = -b/(2a)

Axis of Symmetry of Quadratic  Determine the equation for the axis of symmetry for our equations: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 h(x) = x 2 + 6x - 7  The equation is x = -3  Draw a vertical line through x = -3  Calculate the distance on the x-axis from the axis of symmetry to each zero.  Explain what you notice.

Axis of Symmetry of Quadratic  Can we use the same equation to determine the axis of symmetry for functions that do not cross the x-axis? Graph and determine the equation for the axis of symmetry for: h(x) = x 2 + 6x + 12  Use the line y = 7 to determine the distance from the axis to the function.  Explain your results.

Axis of Symmetry of Quadratic  The axis of symmetry can still be determined by x = (0 –b/a)/2 = -b/(2a) even for functions that do not cross the x-axis.

New Graph  Graph and find axis of symmetry: i(x) = -x 2 + 6x – 8 (a new function)

Location of the Vertex  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8 (the new function)  Explain how can we find the coordinates of the vertices?  Determine the general equation for the coordinates of the vertices.  Vertex is at (-b/2a, f(-b/2a))

Vertex & Axis of Symmetry Summary  Put equation in standard form f(x) = ax 2 + bx + c  Determine the value “a” and “b”  Determine if the graph opens up (a > 0) or down (a < 0)  Find the axis of symmetry:  Find the vertex by substituting the “x” into the function and solving for “y”  Determine two more points on the same side of the axis of symmetry  Graph the axis of symmetry, vertex, & points

Practice: Graphing, Vertex, Axis of Symmetry  Page 58, # 1 – 4 all,  Page 59, # 23 – 34 all  (do some in class together)

End Conditions, Max/Min  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8 (a new function)  What are their end conditions?  Do they have a maximum or minimum?  Explain how we can tell the end conditions and if a function has a maximum or minimum from looking at the equation.

Domain of a Quadratic Function  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8  What is the domain of each equation?  What general rule can we make about the domain of a quadratic function  The domain of a quadratic equation is all real numbers

Range of a Quadratic Function  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8  What is the range (values of y) of each equation?  Does the range differ whether a is positive or negative?  What general rule can we make about the range of a quadratic function?

Range of a Quadratic Function  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8  If a is positive, the range is: y = {y | y  f(-b/2a)}  If a is negative, the range is: y = {y | y  f(-b/2a)}

Practice: Graphing, Vertex, Axis of Symmetry, Min/Max, Open Up/Down, Domain & Range  Page 58 & 59, # 5 – 22 all  Page 59, # 35 – 41 all

Intervals of Increasing and Decreasing  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8  Over what intervals are the functions increasing?  Over what intervals are the functions decreasing?  Explain how the sign of a affects the rules of increasing and decreasing.

Rates of Change (3.3)  Look at our graphs & equations again: f(x) = x 2 + 6x + 5 g(x) = x 2 + 6x + 9 i(x) = -x 2 + 6x – 8  Slope of a linear function is defined as rise/run = (y 2 – y 1 )/(x 2 – x 1 )  These functions are not linear. How can we talk about the slope of these functions?  Explain how the slope of the functions change as we move across the domain.

Practice Rate of Change  Pg 72, # ?? - ??

Summary  For all quadratics:  Axis of symmetry is at x = -b/2a  Vertex is at (-b/2a, f(-b/2a)  The vertex is the extreme  Domain (x) is all real numbers  The zeros, intercepts, solutions, are the determined by moving everything to one side of the equation (equal zero), factoring, and solving via the zero product rule.

Summary  If a > 0  Parabola opens up  Vertex is at the minimum  Rise to the left and right  Range (y) is all real numbers  -b/2a  Rate of change is zero at the vertex, and becomes more negative as x decreases, and more positive as x increases  Intervals of increasing x  vertex  Intervals of decreasing x  vertex

Summary  If a 0)  Parabola opens down  Vertex is at the maximum  Fall to the left and right  Range (y) is all real numbers  -b/2a  Rate of change is zero at the vertex, and becomes more negative as x increases, and more positive as x decreases  Intervals of increasing x  vertex  Intervals of decreasing x  vertex