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Integrated 2 4-1 Graphing Quadratic Functions1 4-1 Graphing Quadratic Functions Warm-up 1.Evaluate the expression 2.Find the value of x in the equation 3.Find the value of y in the equation 4.Find the value of y in the equation 5.Find the approximate value of y to two decimal places in the equation 1.–3 2.x = 1 3.y = 5 4.y = -3/4 5.y ≈ 3.47

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Integrated 2 4-1 Graphing Quadratic Functions2 6-1 Graphing Quadratic Functions Today we will: 1.Understand how the coefficients of a quadratic function influence its graph a.The direction it opens (up or down) b.Its vertex c.Its line of symmetry d.Its y-intercepts Tomorrow we will: 1.Explore translations of parabolas.

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Integrated 2 4-1 Graphing Quadratic Functions3 Parabolas Examples The path of a jump shot as the ball travels toward the basket is a parabola.

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Integrated 2 4-1 Graphing Quadratic Functions4 Key terms Parabola – a curve that can be modeled with a quadratic function. Quadratic function – a function that can be written in the form Standard form of a quadratic function – the form

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Integrated 2 4-1 Graphing Quadratic Functions5 Key terms - continued Vertex – the point where a parabola crosses its line of symmetry. Maximum – the vertex of a parabola that opens downward. The y- coordinate of the vertex is the maximum value of the function. Minimum – the vertex of a parabola that opens upward. The y-coordinate of the vertex is the minimum value of the function. y-intercept – the y-coordinate of the point where a graph crosses the y-axis. x-intercept – the x-coordinate of the point where a graph crosses the x-axis. Line of symmetry

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Integrated 2 4-1 Graphing Quadratic Functions6 Direction and Min/Max If a is positive othe graph opens up othe vertex is a minimum If a is negative othe graph opens down othe vertex is a maximum The graph of the quadratic function, is a parabola.

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Integrated 2 4-1 Graphing Quadratic Functions7 Line of Symmetry and Vertex The line of symmetry is the vertical line. The x-coordinate of the vertex is. To find the y-coordinate of the vertex, substitute for x in the function and solve for y. The y-intercept of the graph of a quadratic function is c.

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Integrated 2 4-1 Graphing Quadratic Functions8 Example 1 Choose the function that models the parabola at the right. A. B. C. D. E.

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Integrated 2 4-1 Graphing Quadratic Functions9 Example 1 Solution The graph opens down so a is negative. B & E are out. The y-intercept is –3. A is out. Find the line of symmetry. Choice C: Choice D: The line of symmetry is x = 4. C is the correct function.

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Integrated 2 4-1 Graphing Quadratic Functions10 Example 2 Use the function A.Tell whether the graph opens up or down. B.Tell whether the vertex is a maximum or a minimum. C.Find an equation for the line of symmetry. D.Find the coordinates of the vertex.

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Integrated 2 4-1 Graphing Quadratic Functions11 Example 2 Solution Use the function A. a is positive, so the graph opens up. B.The vertex is a minimum. C.Equation for the line of symmetry. D.Coordinates of the vertex.

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Integrated 2 4-1 Graphing Quadratic Functions12 Example 3 Use the quadratic function A.Without graphing, will the graph open up or down? B.Is the vertex a minimum or a maximum? C.What is the equation of the line of symmetry? D.Find the coordinates of the vertex of the graph. E.Find the y-intercept. F.Graph the function.

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Integrated 2 4-1 Graphing Quadratic Functions13 Example 3 Solution Use the quadratic function A.The graph will open up, a is positive. B.The vertex a minimum. C.Equation of the line of symmetry. D.Coordinates of the vertex of the graph. E.The y-intercept is y = 25. F.Graph the function.

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Integrated 2 4-1 Graphing Quadratic Functions14 Example 3 Solution Use the quadratic function F.Graph the function. y = 25 line of symmetry x = 3

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Integrated 2 4-1 Graphing Quadratic Functions15 Example 4 Use the function A.Find the y-intercept of the graph. B.Use a graph to estimate the x-intercepts. C.Check one x-intercept by substitution.

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Integrated 2 4-1 Graphing Quadratic Functions16 Example 4 Solution Use the function Solution A.The y-intercept is c or –7.75 B.The x-intercepts are 2.5 and –3.1 C.Check: Substitute 2.5 for x in the original equation.

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Integrated 2 4-1 Graphing Quadratic Functions17 Example 5 Match each equation with its graph. 1 2 3 4

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Integrated 2 4-1 Graphing Quadratic Functions18 Website http://www.valleyview.k12.oh.us/vvhs/dept/ma th/quadshelp.htmlhttp://www.valleyview.k12.oh.us/vvhs/dept/ma th/quadshelp.html

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