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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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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 ________________. 2. For quadratic functions in the form the graph will open up if _________________ and the graph will open down 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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**7. The x-intercepts of a parabola are and**

Determine the x-coordinate of the vertex. 8. 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**

Algebraically Numerically 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.

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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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**Complete the table, plot the points on the graph, and connect**

these points with a smooth parabolic curve. Then complete the missing information. 13. Open upward or downward? Vertex: x-intercepts: y-intercept: Domain: Range: Table Graph

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**Complete the table, plot the points on the graph, and connect**

these points with a smooth parabolic curve. Then complete the missing information. 14. Open upward or downward? Vertex: x-intercepts: y-intercept: Domain: Range: Table Graph

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**Sketching the Graph of a Quadratic Function**

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.

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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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**17. gives the profit in dollars when x units are **

produced and sold. Use the graph of the profit function to determine the following: 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. Profit in $

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**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. (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. L = __________________ x L RIVER

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**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. (b) Express the total area of the fenced corral as a function of x. Hint: Area = (Length)(Width) __________________ x L RIVER

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**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. (c) What is the maximum area that can be enclosed with this fencing? Maximum area = __________________ x L RIVER

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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? Height (ft) Time (sec)

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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 for each graph. See Calculator Perspective 20. 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 for each graph. See Calculator Perspective 21. 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.**

(b) (c)

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**23. Use the given graph to determine the missing input**

and output values. (a) (b)

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

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