- Slides: 28
Chapter 7: Basic Concepts of Algebra Chapter 8: Graphs and Functions
7. 1 Linear Equations • An equation in the variable x is linear if it can be written in the form ax + b = c where a, b, c are real numbers and a is not 0.
7. 1 Helpful properties • Addition property: a = b and a + c = b + c are equivalent (same solutions). • Multiplication property: a = b and ac = bc Are equivalent as long as c is not 0.
7. 1 Solving a linear equation 1. 2. 3. 4. Clear fractions Simplify each side separately Isolate the variable terms on one side Transform so that the coefficient of the variable is 1 5. Check your solution
7. 1 Kinds of Linear equations • Conditional: finite number of solutions Ex. 2 x = 4 • Contradiction: no solutions Ex. 2 x + 1 = 2 x +5 • Identity: true for any number Ex. 2 x + 2 = 2(x + 1)
7. 2 Applications of Linear Equations Key Words Operation Sum, more than, plus, added, increased Addition Less than, difference, minus, decreased Subtraction Times, multiplied by, product Multiplication Quotient, ratio, divided by Division
7. 2 Solving an Applied Problem 1. Read the problem carefully 2. Assign a variable to the unknown value, and write down any other unknowns in terms of this variable. Use tables, diagrams, etc. 3. Write equation using the variable. 4. Solve the equation. 5. State the answer. Is it reasonable? 6. Check the answer in the words of the original problem.
7. 2 Examples # 17: If a quotient of a number and 6 is added to twice the number, the result is 8 less than the number
7. 2 #22 Concert Revenue U 2 generated the top revenue on the concert circuit in 2001. U 2 and second place *NSYNC together took in $196. 5 million in ticket sales. If *NSYNC took in $22. 9 million less than U 2, how much did each generate?
7. 2 #38 Alcohol Mixture How many liters of a 10% alcohol solution must be mixed with 40 liters of a 50% solution to get a 40% solution?
7. 2 #55 Coin Mixture Dave collects US gold coins. He has a collection of 80 coins. Some are $10 coins and some are $20 coins. If the face value of the coins is $1060, how many of each denomination does he have?
7. 2 #76 Time Traveled on a Visit Steve leaves Nashville to visit his cousin Dave in Napa, 80 miles away. He travels at an average speed of 50 miles per hour. One half-hour later Dave leaves to visit Steve, traveling at an average speed of 60 miles per hour. How long after Dave leaves will they meet?
8. 1 Rectangular Coordinate System
8. 1 Rectangular Coordinate System
8. 1 Distance and Midpoint Formulas • The distance between the points (x 1, y 1) and (x 2, y 2) is d = √ (x 2 -x 1)2 + (y 2 -y 1)2 • The midpoint is ((x 1+x 2)/2, (y 1+y 2)/2)
Lines and Slopes 8. 2 • Equations of the form Ax + By = C can be visualized as a straight line • Slope is rise/run • x-intercept: set y = 0 • y-intercept: set x = 0
8. 2 & 8. 3 Equations of Straight Lines • Given the slope m and the y-intercept b, the slope-intercept form is y = mx + b • Given a point (x 1, y 1) and the slope m, the point-slope form is y-y 1 = m(x-x 1)
8. 2 Parallel and Perpendicular • Parallel lines have the same slope Ex: y = 2 x + 1 and y = 2 x – 4 • Perpendicular lines have slopes that are negative reciprocals Ex: y = 2 x + 1 and y = -(1/2)x +3
Functions 8. 4 • A relation is a set of ordered pairs • A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component • Graph of a function obeys the vertical line test: any vertical line crosses at most once
Domain and Range 8. 4 • When ordered pairs are of the form (x, y), x is the independent variable and y is the dependent variable • The domain is the set of all values of the independent variable x • The range is the set of all values of the dependent variable y
Linear Functions • A function that can be written in the form f(x) = mx + b for real numbers m and b is a linear function. • Example: cost and revenue models 8. 4
Quadratic Equations • An equation of the form ax 2 + bx + c = 0 where a, b, c are real numbers with a not equal to 0, is a quadratic equation. • Zero factor property: If ab = 0, then a = 0 or both 7. 7
Applications: 7. 7 #50 7. 7 • The Toronto Dominion Centre in Winnipeg, Manitoba is 407 feet high. Suppose that a ball is projected upward from the top of the center and its position s in feet above the ground is given by the equation s = -16 t 2 + 75 t + 407, where t is the number of seconds elapsed. How long will it take for the ball to reach a height of 450 feet?
7. 7 #66 • A club swimming pool is 30 feet wide by 40 feet long. The club members want a border in a strip of uniform width around the pool. They have enough material for 296 square feet. How wide can the strip be?
Quadratic functions 8. 5 • A function f is a quadratic function if f(x) = ax 2 + bx + c where a, b, and c are real numbers with a not equal to 0.
8. 5 Graphing Quadratic Functions • The graph of the quadratic function defined by f(x) = a(x-h)2 + k, a not 0, is a parabola with vertex (h, k) and the vertical line x = h as axis of symmetry • The graph opens up if a is positive and down if a is negative • The graph is wide if |a|<1 and narrow if |a|>1 compared to y = x 2
More Graphing Quadratics f(x) = ax 2 + bx + c 1. 2. 3. 4. 5. Decide if graph opens up or down Find y-intercept by setting x = 0 Find x-intercept by solving f(x) = 0 Find vertex: x = -b/(2 a) Complete the graph 8. 5
8. 5 #47 • Gina has 100 meters of fencing material to enclose a rectangular exercise run for her dog. What width will give the enclosure the maximum area?