171S3.3p Analyzing Graphs of Quadratic Functions
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value
This graphs either one or two quadratic functions; shows the graph, equation, roots, axis of symmetry, and vertex. Graph on left. http://cfcc.edu/mathlab/geogebra/quadratic_graph_rootsa.html
October 04, 2012
3.3 Analyzing Graphs of
Quadratic Functions
• Find the vertex, the axis of symmetry, and the maximum or minimum value of a quadratic function using the method of completing the square.
• Graph quadratic functions.
• Solve applied problems involving maximum and minimum function values.
View PowerPoint presentation of Quadratic Function.
http://cfcc.edu/faculty/cmoore/QuadraticFunction.ppt
Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
Section 3.3 Graphs of Quadratic Functions Section 3.3 Application: Height of a Baseball
This graphs one quadratic functions; shows the graph, roots, equation, axis of symmetry, and vertex. Graph on right. http://
cfcc.edu/mathlab/geogebra/quadratic_graph_roots.html
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Graphing Quadratic Functions of the Type f (x) = a(x ­ h)2 + k
The graph of a quadratic function is called a parabola. The point (h, k) at which the graph turns is called the vertex. The maximum or minimum value of f(x) occurs at the vertex. Each graph has a line x = h that is called the axis of symmetry.
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Example
Find the vertex, the axis of symmetry, and the maximum or minimum value of f (x) = x2 + 10x + 23.
Solution: Complete the square. Vertex: (–5, –2) Axis of symmetry: x = –5, the x­value at vertex.
Minimum value of the function: ­2, the y­value at vertex.
Login to SAS Curriculum Pathways. Use user name able7oxygen and enter 1445 in the QL # window in the upper right corner. http://sasinschools.com/
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Graph f (x) = x2 + 10x + 23
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Example
Find the vertex, the axis of symmetry, and the maximum or minimum value of
Vertex of a Parabola
Solution: Complete the square.
The vertex of the graph of f (x) = ax2 + bx + c is We calculate the x­coordinate.
Graph:
We substitute to find the y­coordinate.
Vertex: (4, 0)
Axis of symmetry: x = 4
Minimum value of the function: 0
The graph of g is a vertical shrinking of the graph of y = x2 along with a shift of 4 units to the right.
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171S3.3p Analyzing Graphs of Quadratic Functions
October 04, 2012
Application ­ Example
Example
2
For the function f(x) = −x + 14x − 47:
a) Find the vertex.
b) Determine whether there is a maximum or minimum value and find that value.
c) Find the range.
d) On what intervals is the function increasing? decreasing?
Solution
a) f (x) = ­x2 + 14x ­ 47
The x­coordinate of the vertex is:
Since f (7) = ­(7)2 + 14 (7) ­ 47 = 2,
the vertex is (7, 2).
b) Since a is negative (a = –1), the graph opens down, so the second coordinate of the vertex, 2, is the maximum value of the function.
c) The range is (−4, 2].
d) Since the graph opens down, function values increase as we approach the vertex from the left and decrease as we move to the right of the vertex. Thus the function is increasing on the interval (­∞, 7) and decreasing on (7, ∞).
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270/6. Given the quadratic function g(x) = x2 ­ 5x + 6. (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or minimum value and find that value; and (d) graph the function.
Sep 29­12:11 PM
270/14. Given the quadratic function f(x) = ­x2 ­ 8x + 5. (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or minimum value and find that value; and (d) graph the function.
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A stonemason has enough stones to enclose a rectangular patio with 60 ft of stone wall. If the house forms one side of the rectangle, what is the maximum area that the mason can enclose? What should the dimensions of the patio be in order to yield this area?
1. Familiarize. Make a drawing of the situation, using w to represent the width of the fencing.
2. Translate. Since the area of a rectangle is given by length times width, we have
A(w) = (60 ­ 2w)w
= ­ 2w2 + 60w.
3. Carry out. We need to find the maximum value of A(w) and find the dimensions for which that maximum occurs. The maximum will occur at the vertex of the parabola, the first coordinate is
Thus, if w = 15 ft, then the length l = 60 ­ 2 • 15 = 30 ft and the area is 15 • 30 = 450 ft2.
4. Check. (15 + 15 + 30) = 60 feet of fencing.
5. State. The maximum possible area is 450 ft2 when the patio is 15 feet wide and 30 feet long.
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270/10. Given the quadratic function g(x) = (1/3)x2 ­ 2x + 1. (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or minimum value and find that value; and (d) graph the function.
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270/16. Given the quadratic function f(x) = ­3x2 ­ 3x + 1. (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or minimum value and find that value; and (d) graph the function.
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171S3.3p Analyzing Graphs of Quadratic Functions
270/18. Match the equation y = ­(x ­ 4)2 + 3 with one of the graphs (a) ­ (h).
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270/22. Match the equation y = (x ­ 3)2 with one of the graphs (a) ­ (h).
270/24. Match the equation y = 2(x ­ 1)2 ­ 4 with one of the graphs (a) ­ (h).
270/20. Match the equation y = x2 ­ 3 with one of the graphs (a) ­ (h).
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271/26. Determine whether the statement is true or false: The vertex of the graph of f(x) = ax2 + bx + c is ­b / 2a.
271/27. Determine whether the statement is true or false: The graph of h(x) = (x + 2)2 can be obtained by translating the graph of h(x) = x2 right 2 units.
a) Find the vertex. b) Determine whether there is a maximum or minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
271/34. f(x) = (1/2)x2 ­ 3x + 5/2 271/28. Determine whether the statement is true or false: The vertex of the graph of the function g(x) = 2(x ­ 4)2 ­ 1 is (­4, ­1).
271/30. Determine whether the statement is true or false: The minimum value of the function f(x) = 3(x ­ 1)2 + 5 is 5.
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a) Find the vertex. b) Determine whether there is a maximum or minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
271/36. f(x) = ­2x2 ­ 24x ­ 64 Sep 29­12:36 PM
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a) Find the vertex. b) Determine whether there is a maximum or minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
271/38. f(x) = ­3x2 + 24x ­ 49 Sep 29­12:36 PM
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171S3.3p Analyzing Graphs of Quadratic Functions
a) Find the vertex. b) Determine whether there is a maximum or minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
271/40. g(x) = 2x2 ­ 6x + 5 October 04, 2012
271/42. Height of a Projectile. A stone is thrown directly upward from a height of 30 ft with an initial velocity of 60 ft/sec. The height of the stone t seconds after it has been thrown is given by the function s(t) = ­16t2 + 60t + 30. Determine the time at which the stone reaches its maximum height and find the maximum height.
(a) Vertex: (1.5, 0.5)
(b) a = 2 > 0, curve opens upward and has a minimum value of 0.5
(c) Range is y > 0.5 or [0.5, ∞).
(d) Increasing: (0.5, ∞); decreasing (­∞, 0.5)
http://cfcc.edu/mathlab/geogebra/quadratic_graph_rootsa.html
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271/44. Height of a Rocket. A model rocket is launched with an initial Sep 29­12:40 PM
271/46. Maximizing Area. A fourth­grade class decides to enclose a rectangular garden, using the side of the school as one side of the rectangle. What is the maximum area that the class can enclose with 32 ft of fence? What should the dimensions of the garden be in order to yield this area?
velocity of 150 ft / sec from a height of 40 ft. The function s(t) = ­16t2 + 150t + 40 gives the height of the rocket, in feet, t seconds after it has been launched. Determine the time at which the rocket reaches its maximum height and find the maximum height. Sep 29­12:40 PM
272/48. Maximizing Area. The sum of the base and the height of a parallelogram is 69 cm. Find the dimensions for which the area is a maximum.
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272/50. Maximizing Profit. In business, profit is the difference between revenue and cost; that is,
Total profit = Total revenue ­ Total cost or P(x) = R(x) ­ C(x),
where x is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for R(x) = 5x and C(x) = 0.001x2 + 1.2x + 60
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171S3.3p Analyzing Graphs of Quadratic Functions
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272/52. Maximizing Profit. In business, profit is the difference between revenue and cost; that is,
Total profit = Total revenue ­ Total cost or P(x) = R(x) ­ C(x),
where x is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for R(x) = 20x ­ 0.1x2 and C(x) = 4x + 2
P(x) = R(x) - C(x)
P(x) = (20x - 0.1x2) - (4x + 2)
P(x) = -0.1x2 + 16x - 2
Vertex: (80, 638)
Maximum profit is 638 (units of
money) when 80 units are sold.
http://cfcc.edu/mathlab/geogebra/quadratic_graph_rootsa.html
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