Sections 3.5, : Quadratic Functions

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1 Week 7 Handout MAC 1105 Professor Niraj Wagh J Sections 3.5, : Quadratic Functions A function that can be written in the form f(x)= ax 2 +bx+c for real numbers a, b, and c, with a not equal to zero, is called a quadratic function in standard form. The Vertex Form of a Quadratic Function is f(x)=a(x-h) 2 + k. If a > 0 then you have an upward parabola or a J (YAY!) If a < 0 then you have a downward parabola or a L (WAH!) The vertex is (h, k) The max or min is f(h)= k Where h and k are computed by completing the square. The Vertex More specifically, to find the vertex (the highest or lowest point of a parabola) we first find the x coordinate (axis of symmetry) by computing x = b. Then after 2a you find the x coordinate you plug it back into the function to solve for y. Essentially # Vertex = b 2a, f # b& & % % (( $ $ 2a' ' The Y-coordinate is the actual MAX or MIN of the function. N. Wagh 1

2 Variation 1: Vertical Shifts The graph of y= f(x) + k moves k units up on the y-axis. The graph of y=f(x) k moves k units down on the y-axis. EXAMPLE: Find the vertex and then graph y = x N. Wagh 2

3 Variation 2: Horizontal Shifts The graph of y= f(x-h) moves h units to the right. The graph of y= f(x+h) moves h units to the left. EXAMPLE: Find the vertex and then graph y = (x-2) N. Wagh 3

4 Variation 3: Reflection over axes The graph y= -f(x) reflects the graph y=f(x) over the x-axis. The graph y= f(-x) reflects the graph y=f(x) over the y-axis. For example: y = -(x) 2 means you reflect over the x-axis. y = (-x) 2 means you would reflect over the y-axis. EXAMPLE: Find the vertex and then graph y = -x 2 N. Wagh 4

5 Variation 4: Vertical compressing/stretching over graphs The graph of y= af(x) If a>1 stretch the graph of y=f(x) vertically by a factor of a. If 0<a<1, shrink the graph of y=f(x) vertically by a factor of a. -> Multiply each y-coordinate of y = f(x) by a. Variation 5: Horizontal compressing/stretching over graphs The graph of y= f(ax) If a>1 shrink the graph of y= f(x) horizontally by a factor of 1/a. If 0<a<1 stretch the graph of y=f(x) horizontally by a factor of 1/a. EXAMPLE: Find the vertex and then graph y = 5x 2 N. Wagh 5

6 PRACTICE. PRACTICE. PRACTICE. Find the vertex and then graph the following functions. 1. y = (x+1) y = 2(x-1) N. Wagh 6

7 REVIEW: To convert from standard form to vertex form, you need to complete the square. To go the other direction, you simplify (by distributing) Examples OPTION 1 METHOD: (a) Write the following functions in vertex form, i.e. f(x) = a(x-h) 2 + k. (b) Graph the functions by starting with the graph f(x) = x 2 and using transformations (shifting, compressing, stretching, and/or reflection). 1. f(x) = 3x 2-18x + 5 N. Wagh 7

8 OPTION 2 METHOD: (a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and range of the function. (c) Determine where the function is increasing and where it is decreasing. 2. f (x) = x 2 2x 3 3. Determine the quadratic function with a vertex of (2, 1) and y-intercept of (0, 5). N. Wagh 8

9 Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find its value. 4. y = 2 3 x2 +12x 3 5. The price p (in dollars) and the quantity x sold of a certain product obey the demand equation: p = 1 6 x +100 (a) Find a model that represents the revenue R as a function of x. Hint: Revenue = Price * Quantity (b) What is the domain of R? (c) What is the revenue if 200 units are sold? (d) What quantity x maximizes revenue? What is the maximum revenue? (e) What price should the company charge to maximize revenue? N. Wagh 9

10 PRACTICE. PRACTICE. PRACTICE. Write y = 2x 2 4x + 5 in vertex form. Find the vertex. Graph the parabola and the maximum or minimum. Fantastic! We ve completed the examples for this section! Now work on HW 4.3 & 4.4 in MyMathLab. If you have any questions, please let me know! I will be more than happy to help you! J N. Wagh 10

Quadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0

Quadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0 Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,