Graphing Quadratic Functions

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

Graphing Quadratics If 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 plane If the parabola points up or down, and whether it opens normal, narrow or wide Our graphs will be more “quick sketches” than exact graphs.

Graph of f(x)=x2 Axis is x = 0 x f(x) 1 -1 2 -2 4 1 -1 2 -2 4 Points up, opens “normal” Notice the symmetry Vertex at (0, 0)

More with Vertex Form The 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 = h If a > 0, the graph points up If a < 0, the graph points down Example #2 Example #1 Vertex is (4, 0) Axis is x = 4 Points down Vertex is _____ Axis is _______ Points _______ Vertex is (0, 6) Axis is x = 0 Points up Vertex is _____ Axis is _______ Points _______ Example #3 Vertex is (-3, -1) Axis is x = -3 Points up Vertex is _____ Axis is _______ Points _______ Notice that you take the opposite of h from how it is written in the equation

Equations of Quadratic Functions Vertex Form Standard 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 If a > 0, the graph points up If a < 0, the graph points down Example #1 b = 4, a = -1 Find x-value of vertex using formula Vertex is _____ Axis is _______ Points _______ Vertex is (2, 1) Axis is x = 2 Points down Find y-value using substitution 2 (2) (2)

More examples Example #2 You try: Vertex is _____ Axis is _______ Points _______ Find x-value using formula b = -1, a = 3 Find y-value using substitution 16 16 16 Vertex is (3, 17) Axis is x = 3 Points down Vertex is _____ Axis is _______ Points _______ Vertex is (1/6, 59/12) Axis is x = 1/6 Points up

More 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 wider If a is farther from 0, the graph opens narrower If a > 0, the graph points up If a < 0, the graph points down

Graphing Quadratics If 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 plane The value of a, because we need to know if it points up or down, and whether it opens normal, narrow or wide Our graphs will be more “quick sketches” than exact graphs.

Sketch each quadratic V = (-3, -1) V = (2, 1) Points up Points down Narrow V = (2, 1) Points down Normal V = (-4, 2) Points up Normal V = (0, 4) Points down Wide

Finding x-intercepts of quadratic functions What are other words for x-intercepts? Name 4 methods of finding the x-intercepts of quadratic equations: roots zeroes solutions All are the value of x when y = 0 factoring The Square Root The Quadratic Formula Graphing

Summary: Be able to compare and contrast vertex and standard form   Vertex Form Standard 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? 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)

Max and Min Problems 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 max min If a quadratic points down, the vertex is a maximum point If a quadratic points up, the vertex is a minimum point If you are asked to find a maximum or minimum value of a quadratic function, all you need to do is find its vertex

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) = -16t2 + 8t + 100. 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?

vertex 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 vertex

Example f(t) = -16t2 + 8t + 100. f(1/4) = -16(1/4)2 + 8(1/4) + 100. Step 2: Understand the equation Example y x Input: time Output: height Step 1: Visualize the problem f(t) = -16t2 + 8t + 100. To find the max values, find the vertex The x-value of the vertex is the max time (1/4, 101) It took about .25 seconds for the object to reach its max height The y-value of the vertex is the max height f(1/4) = -16(1/4)2 + 8(1/4) + 100. f(t) = -16t2 + 8t + 100. The max height was 101 feet f(1/4) = 101