SECTION 3.1: Quadratic Functions

Transcription

1 SECTION 3.: Quadratic Functions Objectives Graph and Analyze Quadratic Functions in Standard and Verte Form Identify the Verte, Ais of Symmetry, and Intercepts of a Quadratic Function Find the Maimum or Minimum of a Quadratic Function Build Quadratic Models from Verbal Descriptions I. Quadratic Functions in Standard Form and Verte Form QUADRATIC FUNCTIONS Standard Form f() = a + b + c Parabola where a, b, c are real numbers and a 0. Verte (Transformation) Form f() = a( h) + k Parabola y = vertically stretched or compressed by a and shifted horizontally h units and vertically k units. The graph opens up if a > 0 and down if a < 0 Verte is the minimum or maimum point on the parabola ( b, f ( b )) a a Verte is the minimum or maimum point on the parabola (h, k) If the parabola opens up, the function has a minimum value. If the parabola opens down, the function has a maimum value. Maimum and minimum values occur at the verte. The function s minimum or maimum value is the y-value of the verte and it occurs at the -value of the verte. The ais of symmetry is the vertical line = b a The ais of symmetry is the vertical line = h. Domain: All Reals (, ) Range: [f ( b ), ) if parabola opens up; a (, f ( b )] if parabola opens down. a -intercepts (zeros): Set f() = 0 and solve for. Range: [k, ) if parabola opens up; (, k] if parabola opens down y-intercept: Find f(0). That is, set = 0 and solve for f().

2 E. f() = 3( + 4) + 3 a. The given function is the parent function with the following transformations applied: b. The function is given in form. c. Rewrite the function in standard form. d. The function opens. e. The verte of the function is. f. The function has a of that occurs at. g. The ais of symmetry is. h. Find the zeros of the function, if any y i. Find the y-intercept.

3 E. f() = + 3 a. The function is in what form? b. How does the parabola open? c. Verte (maimum or minimum?) d. -intercept(s) e. y-intercept f. Ais of symmetry g. Domain: h. Range: i. Increasing: j. Decreasing: y f() = + 3 3

4 E3. f() = ( ) a. The function is in what form? b. How does the parabola open? c. Verte (ma or min?) d. -intercept(s) e. y-intercept f. Ais of symmetry g. Domain: h. Range: i. Increasing: j. Decreasing: y f() = ( ) 4

5 E4. Find the equation of the quadratic function given the verte ( 3, 5 ) and passing through ( 4, ). Then find the zeros of the function. II. Applications E5. Minimizing Unit Cost. A small business makes Tie Dye T-shirts. The unit cost C (the cost in dollars to make each T-shirt) depends on the number of T-shirts made. After months of data collection, the sales team determines that if T-shirts are made, the unit cost is approimated by the function: C() = a. How many Tie Dye T-Shirts should be made to minimize the unit cost? b. What is the minimum unit cost? 5

6 E6. Maimizing Revenue The price p (in dollars) and the quantity sold of a vacuum cleaner obey the demand equation: p = a. Find the model that epresses the revenue R as a function of. (remember R = p) b. What quantity maimizes revenue? c. What is the maimum revenue? d. What price p should the company charge to maimize revenue? E7. Maimizing Height. Andrew throws a ball vertically upward. After t seconds, its height h (in feet) is given by the function h(t) = 84t 6t. a. When will the ball reach its maimum height? b. What is the maimum height that the ball will reach? c. When will Andrew catch the ball? 6

7 E8. Optimizing Area. A school wishes to form three sides of a rectangular playground using 0 meters of fencing. The playground borders the school building, so the fourth side of the playground does not need fencing. a. Find a function that gives the area A() of the playground (in square meters) in terms of. b. What value of gives the maimum area that the playground can have? c. What is the maimum area that the playground can have? E9. Optimizing Area. A wire is 48 inches long. The wire is cut into two pieces, and each piece is bent and formed into the shape of a square. Suppose that the side length (in inches) of one square is. a. Find a function that gives the area A() enclosed by the two squares (in square inches) in terms of. 48 inches b. Find the side length that minimizes the total area of the two squares. c. What is the minimum area enclosed by the two squares? 7