25 Quadratic Functions Graph is a parabola Either

25. Quadratic Functions Ø Ø Graph is a parabola. Either has a minimum or maximum point. That point is called a vertex. Use transformations on x 2 and -x 2 to get graph of any quadratic function. 1

Example 1 f(x) = x 2: shift up 4 units and shift to the left 5 units 2

Standard Form 3

Example 2 Graph the function: Minimum value is Domain: Range: 4

Example 3 Graph the function: Maximum value is Domain: Range: 5

Example 4 Find the maximum or minimum value of the function. Minimum value is x-intercepts = y-intercepts = 6

x value of the vertex 7

Examples Find the maximum or minimum value of the functions. 8

26. Variation Certain situations exist where: q If one quantity increases, the other quantity also increases. q If one quantity increases, the other quantity decreases. This kind of modeling is called variation. 9

Direct Variation Definition: When one quantity increases, the other increases. When one quantity decreases, the other decreases. 10

Indirect Variation Definition: When one quantity increases, the other decreases. When one quantity decreases, the other increases. 11

Joint Variation Definition: When one quantity varies according to two (or more) other quantities. 12

Example 1 Express the statement as an equation. Use the given information to find the constant of proportionality. y is directly proportional to x. When x = 3, y = 39. 13

Example 2 Express the statement as an equation. Use the given information to find the constant of proportionality. M is inversely proportional to the square of t. When t = -4, M = 5. 14

Example 3 Express the statement as an equation. Use the given information to find the constant of proportionality. C is proportional to cube root of s and inversely proportional to t. When t = 7 and s = 27, C = 66. 15

Example 4 The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. Write an equation that expresses this variation. Find the constant of proportionality if 125 L of gas exerts a pressure of 12. 7 k. Pa at a temperature of 250 K. Find the pressure if volume is increased to 150 L and temperature is decreased to 200 K. (Round the answer to the nearest tenth. ) 16

Example 5 The force F needed to keep a car from skidding is jointly proportional to the weight w of the car and the square of the speed s, and is inversely proportional to the radius of the curve r. Write an equation that expresses this variation. A car weighing 3, 600 lb travels around a curve at 60 mph. The next car to round the curve weighs 2, 500 lb and requires the same force as the first. How fast was the second car traveling? 17

Example 5(continued) First Car(3600 lb, 60 mph) Second Car(2500 lb, ? mph) 18

27. Systems of Equations A set of equations involving the same variables l A solution is a collection of values that makes each equation true. l Solving a system = finding all solutions l 19

Example 1 Is (x, y) = (2, -2) a solution? Is (1, -1/3) a solution? 20

Substitution Method l l l Pick one equation and solve for one variable in terms of the other. Substitute that expression for the variable in the other equation. Solve the new equation for the single variable and use that value to find the value of the remaining variable. 21

Example 2 22

Elimination Method l l l Multiply both equations by constants so that one variable has coefficients that add to zero. Add the equations together to eliminate that variable. Solve the new equation for the single variable and use that value to find the value of the remaining variable. Example 3 23

Example 4 24

28. Systems of Linear Equations l l l A set of linear equations involving the two variables A solution is the intersection of the two lines. One of three things can happen: 25

Example 1 26

Example 2 27

Example 3 A chemist wants to mix a 20% saline solution with a 40% saline solution to get 1 liter of a 26% solution. How much of each should she use? (1 liter = 1000 ml) 28

Example 4 20 miles A boat travels downstream for 20 miles in 1 hour. It turns around and travels 20 miles upstream (against the current) in 1 hours and 40 minutes. What is the boat’s speed and how fast is the current? 29

Example 5 A woman invested in two accounts, one earned 2% and the other earned 10% in simple interest. She put twice as much in the lower-yielding account. If she earned $3500 in interest last year, how much was invested in each account? 30