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1 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 1 L3-Wed-6-Oct-016-Sec-3-1-Lin-Funct-HW3-3-3-Quad-Funct-HW4-Q19
2 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page Sec 3.1 Linear Functions and Their Properties A lot of this is review from our study of lines. Average Rate of Change of a function from a to c is the slope of the secant line through the two points. (A Secant line is a line that intersects a curve, usually at more than one point.) y f c f a ARC a c x c a So, the average rate of change for a linear function is the slope of the line, m. [ mc ( ) b] [ ma ( ) b] ARC c a mc ( ) bma ( ) b c a mc ( ) ma ( ) c a mc ( a) c a m
3 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 3 First, we figure out what the functions are given their graphs: m 1 y x b Use, 6 : to find b f 6 b b x y f x x g x 8 h x 6 x g x For what value of x are the two functions equal? x 8 x 10 x 5 f x f x h x 8 x 6 For what values of x is f between g and h? 10 x 4 5 x g x 5 or, 5
4 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 4 a. What are the equilibrium price and quantity? This is where supply equals demand: Dp S p 3000p p 10, p 1,000 p 3 So, supply will equal demand if you charge $3 per dog. The supply should be S The demand will be D ,
5 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 5 D p S p 1000p 10, p 000 1, p 3 p If the price is over $3 the supply will be greater than the demand. In that case, people will want fewer dogs than you have on hand and so some will be left over and have to be sent to a school lunch program resulting in a loss in your profits. If the price is under $3, the supply will be less than the demand and so people will want more dogs than they can get (you will run out). They will be angry. Also, you will lose profit since you could have sold more than you did.
6 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 6 Sec 3.3 Quadratic Functions and Their Properties
7 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 7 If we begin with the definition of a quadratic function f x ax bx c and complete the square we can write the function in a different, and sometimes more useful, form. We first must factor out the a so the coefficient of the x squared term is 1: f x ax bx c b ax x c a Complete the square by adding and subtracting b f x a x x c a b b b a x x c a 4a 4a b b b a x x a c a 4a 4a b b a x c a 4a b b 4a a x c a 4a 4a b 4ac b a x a 4a 1 b b a 4a
8 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 8 5 Remember Casandra of Greek mythology. The god Apollo gave her the gift of prophecy. Then he got mad at her and wanted to take the gift back but could not. So, he made it that she still could tell the future but no one would believe her. She knew the truth but could not prove it to anyone.
9 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 9 This is the Quadratic Formula. Let s make this formula a bit more simple by making a couple of substitutions:
10 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 10 We could also find the vertex by completing the square: f x x 4x 1 x x 1 x x 111 x 1 x x 1 1 x 1 3 We can see that the vertex is at (1, 3).
11 L3-Wed-6-Oct-016-Sec-3-1-Linear-Funct-HW3-3-3-Quad-Funct-HW4-Q19 page 11 An easy way to memorize the formula for the x component of the vertex is to connect it to the quadratic formula. b b 4ac b b 4ac x1, a a a The b term is the x-component of the lowest (or highest) point on the parabola. The second term a moves us either to the left or right of that point (in other words, away from the vertex). We did this sort of thing when we graphed piecewise defined functions. Now that we know the vertex form of the parabola we can do that faster. f x a x h k From the graph, we see that the vertex is at (, 1). So, we have h =, k = 1: f x a x 1 From the graph, the y-intercept is (0, 5) so we can find a: a a 1 1 f x a x a So, we have f x 1 x 1
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