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Graphing Quadratic Functions

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?.

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Graphing Quadratic Functions

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  1. Graphing Quadratic Functions

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

  3. 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.

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

  5. 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 More with Vertex Form If a > 0, the graph points up If a < 0, the graph points down Example #2 Example #1 Vertex is _____ Axis is _______ Points _______ Vertex is (4, 0) Axis is x = 4 Points down 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

  6. Vertex Form Standard Form Equations of Quadratic Functions

  7. 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 More with Standard Form 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)

  8. Example #2 You try: Vertex is _____ Axis is _______ Points _______ More examples 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

  9. 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

  10. 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.

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

  12. The Square Root 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 Quadratic Formula

  13. Summary: Be able to compare and contrastvertex and standard form

  14. Max and Min Problems The vertex of a quadratic function is either a maximum point or a minimum point What is the definition of the maximum or minimum point of a quadratic function? 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

  15. 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?

  16. Example What was the maximum heightof 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

  17. Step 2: Understand the equation Example y x Output: height Input: time 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

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