24.1 Graphing Quadratic Functions Learning Target: I can Identify and graph quadratic equations.
3Graphs of Quadratic Functions Axis of symmetryImportant features of graphs of parabolasMaximumx-interceptsMinimumVertex
4Graphing QuadraticsIf you were asked to graph a quadratic, what information would you need to know to complete the problem?The vertex, because we need to know where the graph is located in the planeIf the parabola points up or down, and whether it opens normal, narrow or wideOur graphs will be more “quick sketches” than exact graphs.
5Graph of f(x)=x2 Axis of symmetry is x = 0 x f(x) 1 -1 2 -2 4 1-12-24Points up, opens “normal”Notice the symmetryVertex at (0, 0)The “mother function”
6More with Vertex FormThe vertex is (h, k). Changes in (h, k) will shift the quadratic around in the plane (left/right, up/down).The axis of symmetry is x = hIf a > 0, the graph points upIf a < 0, the graph points downExample #2Example #1Vertex is (4, 0)Axis is x = 4Points downVertex is _____Axis is _______Points _______Vertex is (0, 6)Axis is x = 0Points upVertex is _____Axis is _______Points _______Example #3Vertex is (-3, -1)Axis is x = -3Points upVertex is _____Axis is _______Points _______Notice that you take the opposite of h from how it is written in the equation
7Equations of Quadratic Functions Vertex FormStandard Form
8More about a When a = 1, the graph is “normal” a =1 a =1/5 a = 5 What happens to the graph as the value of a changes?If a is close to 0, the graph opens _______________If a is farther from 0, the graph opens ____________If a > 0, the graph points________If a < 0, the graph points ________If a is close to 0, the graph opens widerIf a is farther from 0, the graph opens narrowerIf a > 0, the graph points upIf a < 0, the graph points down
9Max and Min ProblemsWhat 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 pointmaxminIf a quadratic points down, the vertex is a maximum pointIf a quadratic points up, the vertex is a minimum pointIf you are asked to find a maximum or minimum value of a quadratic function, all you need to do is find its vertex
10Sketch each quadraticIdentify the vertex, axis of symmetry, the max or min value, and domain and range.AOS: x = -3 Min: k =-1 D: all reals R: all reals > - 1V = (-3, -1)Points upNarrowAOS: x = 0Max: k = 4D: all realsR: all reals < 4V = (0, 4)Points downWide
11Sketch each quadratic V = (-3, -1) V = (2, 1) Points up Points down NarrowV = (2, 1)Points downNormalV = (-4, 2)Points upNormalV = (0, 4)Points downWide
12HomeworkHomework – p. 199 #9-36 by 3’sChallenge - 56
13Summary: Be able to compare and contrast vertex and standard form Vertex FormStandard FormHow do you find the Vertex?How do you find the Axis of Symmetry?How can you tell if the function:points up or down?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 methodSet = 0 and use method of choice (factor, formula or square root)
14More with Standard Form To find the x-value of the vertex, use the formulaTo find the y-value, plug in x and solve for yThe axis of symmetry isIf a > 0, the graph points upIf a < 0, the graph points downExample #1b = 4, a = -1Find x-value of vertex using formulaVertex is _____Axis is _______Points _______Vertex is (2, 1)Axis is x = 2Points downFind y-value using substitution2(2)(2)
15More examples Example #2 You try: Vertex is _____ Axis is _______ Points _______Find x-value using formulab = -1, a = 3Find y-value using substitution161616Vertex is (3, 17)Axis is x = 3Points downVertex is _____Axis is _______Points _______Vertex is (1/6, 59/12)Axis is x = 1/6Points up
16Finding x-intercepts of quadratic functions What are other words for x-intercepts?Name 4 methods of finding the x-intercepts of quadratic equations:rootszeroessolutionsAll are the value of x when y = 0factoringThe Square RootThe Quadratic FormulaGraphing
17ExampleAn 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) = -16t2 + 8tWhat 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?
18vertex 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 pointvertex
19Example f(t) = -16t2 + 8t + 100. f(1/4) = -16(1/4)2 + 8(1/4) + 100. Step 2: Understand the equationExampleyxInput: timeOutput: heightStep 1: Visualize the problemf(t) = -16t2 + 8tTo find the max values, find the vertexThe x-value of the vertex is the max time(1/4, 101)It took about .25 seconds for the object to reach its max heightThe y-value of the vertex is the max heightf(1/4) = -16(1/4)2 + 8(1/4)f(t) = -16t2 + 8tThe max height was 101 feetf(1/4) = 101