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Section 8.4 Quadratic Functions

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8.4 Lecture Guide: Quadratic Functions Objective 1: Distinguish between linear and quadratic functions. A second-degree polynomial function can be written in the form and is called a quadratic function.

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First Degree Functions --- Linear Functions Algebraically:Numerically:Graphically: A fixed change in x produces a constant change in y. A straight line Example:

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Second Degree Functions --- Quadratic Functions Algebraically:Numerically:Graphically: The y-values from a symmetric pattern about the vertex. A parabola Example:

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2. For quadratic functions in the form the graph will open up if _________________ and the graph will open down if __________________. 1. For a linear function in the form the graph will slope upward to the right if __________________ and the graph will slope downward to the right if ________________.

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Identify the graph of each function as a line or a parabola. If the graph is a line, determine whether the slope is negative or positive. If the graph is a parabola, determine whether the parabola is concave up (the graph opens up) or concave down (the graph opens down). 3.4.

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Identify the graph of each function as a line or a parabola. If the graph is a line, determine whether the slope is negative or positive. If the graph is a parabola, determine whether the parabola is concave up (the graph opens up) or concave down (the graph opens down). 5.6.

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Objective 2: Determine the vertex of a parabola.

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For a parabola defined by, the x-intercepts (if they exist) can be determined by using the quadratic formula. The vertex will be located at the x-value midway between the two x-intercepts. See the figure below. y x

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8. The x-intercepts of a parabola are and Determine the x-coordinate of the vertex. 7. The x-intercepts of a parabola are and Determine the x-coordinate of the vertex.

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Vertex of the Parabola Defined by AlgebraicallyNumerically Example: a = −1 and b = 7 The y-values form a symmetric pattern about the vertex. If the table contains the vertex, the y-coordinate of the vertex will be either the largest or the smallest y-value in the table. Example:

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Graphically The vertex is either the highest or the lowest point on the parabola. Example:

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Finding the Vertex of the Parabola defined by Step 1. Determine the x-coordinate using Step 2. Then evaluate to determine the y-coordinate.

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9. Determine the vertex of the parabola defined by

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10. Determine the vertex of the parabola defined by

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Use the given equation to calculate the x and y-intercepts and the vertex of each parabola. 11. (a) y-intercept (b) x-intercepts (c) Vertex

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Use the given equation to calculate the x and y-intercepts and the vertex of each parabola. 12. (a) y-intercept (b) x-intercepts (c) Vertex

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Objective 3: Sketch the graph of a quadratic function and determine key features of the resulting parabola.

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Open upward or downward? Vertex: x-intercepts: y-intercept: Domain: Range: Table Graph Complete the table, plot the points on the graph, and connect these points with a smooth parabolic curve. Then complete the missing information. 13.

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Open upward or downward? Vertex: x-intercepts: y-intercept: Domain: Range: Table Graph Complete the table, plot the points on the graph, and connect these points with a smooth parabolic curve. Then complete the missing information. 14.

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1. Determine whether the parabola opens upward or downward. 2. Determine the coordinates of the vertex. 3. Determine the intercepts. 4. Complete a table using points on both sides of the vertex. 5. Connect all points with a smooth parabolic shape. Sketching the Graph of a Quadratic Function

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15. Sketch the graph of (a) Will the parabola open upward or open downward? (b) Determine the coordinates of the vertex. (c) Complete a table of values using inputs on both sides of the vertex.

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15. Sketch the graph of (d) Determine the intercepts of the graph of this function. (e) Use this information to sketch the graph of this function.

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16. Sketch the graph of (a) Will the parabola open upward or open downward? (b) Determine the coordinates of the vertex. (c) Complete a table of values using inputs on both sides of the vertex.

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16. Sketch the graph of (d) Determine the intercepts of the graph of this function. (e) Use this information to sketch the graph of this function.

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Objective 4: Solve problems involving a maximum or minimum value.

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Units sold (a) Overhead costs (Hint: Evaluate.) (b) Break even values (Hint: When does ?) (c) Maximum profit that can be made and the number of units to sell to create this profit. 17. gives the profit in dollars when x units are produced and sold. Use the graph of the profit function to determine the following:

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(a) If x yards are used for the two parallel sides, how much fencing remains for the side parallel to the river? Give this length in terms of x. x L RIVER x L = __________________ 18. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

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(b) Express the total area of the fenced corral as a function of x. Hint: Area = (Length)(Width) x L RIVER x __________________ 18. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

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(c) What is the maximum area that can be enclosed with this fencing? x L RIVER x Maximum area = __________________ 18. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

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19. The equation gives the height y of a baseball in feet x seconds after it was hit. (a) Use the equation to determine how many seconds into the flight the maximum height is reached. (b) Determine the maximum height the ball reached.

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19. The equation gives the height y of a baseball in feet x seconds after it was hit. (c) Do your results agree with what you can observe from the graph? Time (sec) Height (ft)

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Use your graphing calculator to determine the minimum/maximum value of and the x-value at which this minimum/maximum occurs. Use a window of by 20. for each graph. See Calculator Perspective Sketch of calculator graph: Max/min value: x-value where max/min occurs:

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Use your graphing calculator to determine the minimum/maximum value of and the x-value at which this minimum/maximum occurs. Use a window of by 21. for each graph. See Calculator Perspective Sketch of calculator graph: Max/min value: x-value where max/min occurs:

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22. Use the function to solve each equation and inequality. (a) (c) (b)

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23. Use the given graph to determine the missing input and output values. (a) (b)

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24. Use the function to determine the missing input and output values. (a) (b)

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