Entry Task

Take a look…. y = x(18-x) Then we had y = -x 2 +18x We could graph this using symmetry and find the zero’s. if x is 0 what is y? 0 or 18. Knowing its symmetrical, we know the x coordinate of the vertex has to be 9 and now we can find the y.

4.2 Standard Form of a Quadratic Equation Target: I can graph functions written in standard form.

Graphs of Quadratic Functions Vertex Axis of symmetry x-intercepts Important features of graphs of parabolas

Equations of Quadratic Functions Vertex FormStandard Form

More with Standard Form To find the x-value of the vertex, use the formula To find the y-value, plug in x and solve for y The axis of symmetry is Example #1 If a > 0, the graph points up If a < 0, the graph points down Vertex is _____ Axis is _______ Points _______ b = 4, a = -1Find x-value of vertex using formula Find y-value using substitution 2(2) Vertex is (2, 1) Axis is x = 2 Points down y = -x 2 + 4x – 3

More examples Example #2 Vertex is _____ Axis is _______ Points _______ b = -1, a = 3Find x-value using formula Find y-value using substitution 1616 Vertex is (1/6, 59/12) Axis is x = 1/6 Points up You try: Vertex is _____ Axis is _______ Points _______ Vertex is (3, 17) Axis is x = 3 Points down

Sketch each quadratic V = (-4, 2) Points up Normal V = (2, 1) Points down Normal Vertex: AOS: Min or Max: Range: Vertex: AOS: Min or Max: Range:

Finding x-intercepts of quadratic functions What are other words for x-intercepts? Name 4 methods of finding the x-intercepts of quadratic equations: All are the value of x when y = 0

Example An object is thrown upward from the top of a 100 foot cliff. Its height in feet about the ground after t seconds given by the function f(t) = -16t 2 + 8t What was the maximum height of the object? How many seconds did it take for the object to reach its max height? How can we find the answer? What is the question asking for?

Example What was the maximum height of the object? How many seconds did it take for the object to reach its maximum height? What is the definition of the maximum or minimum point of a quadratic function? The vertex of a quadratic function is either a maximum point or a minimum point

Example Step 1: Visualize the problem f(t) = -16t 2 + 8t To find the max values, find the vertex The x-value of the vertex is the max time The y-value of the vertex is the max height Output: height Input: time It took about.25 seconds for the object to reach its max height f(t) = -16t 2 + 8t f(1/4) = -16(1/4) 2 + 8(1/4) f(1/4) = 101 The max height was 101 feet (1/4, 101) Step 2: Understand the equation

Standard to Vertex form….. Write y = -x 2 – 4x + 6 in vertex form We know x = 4/2 = 2 So f(2) = (2) + 6 = 2 So the vertex is (2,2) The a is the exact same in standard and vertex form…. f(x) = -(x-2) 2 + 2

Homework Homework– p. 206 # 9-30 by 3’s, #38, Challenge -

Summary: Be able to compare and contrast vertex and standard form Vertex FormStandard Form How do you find the Vertex? How do you find the Axis of Symmetry? How can you tell if the function: points up or down? points up or down? opens normal, wide or narrow?opens normal, wide or narrow? What info is needed to do a quick sketch or graph? How do you find the solutions? (x-intercepts, roots, zeroes, value of x when y = 0) Set = 0, get “squared stuff” alone, then use square root method Set = 0 and use method of choice (factor, formula or square root)