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Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope and y-intercept: a) y = 3x + 1 b) 2x + 3y = 1 Ex. Graph: a) y 21 x 4 2 -5 5 -2 -4 Ex. Graph: b) y2 4 2 -5 5 -2 -4 Ex. Graph: c) xy 2 4 2 -5 5 -2 -4 Slope Thm. The slope between (x1,y1) and (x2,y2) is y2 y1 m x2 x1 Ex. Find the slope between (-2,0) and (3,1). • Rising line has positive slope: 2 5 ,4 3 8 , • Falling line has negative slope: 7 • Horizontal line as slope of 0: 0 0 5 3 , • Vertical line has undefined slope: 2 8 0 0 , A vertical line has an equation like x = 3, and can’t be written as y = mx + b Thm. Parallel lines have the same slope. Thm. Perpendicular lines have slopes that are negative reciprocals. Ex. Are the lines parallel, perpendicular, or neither? (3,-1) to (-3,1) and (0,3) to (-1,0) Thm. A line with m = slope that passes through the point (x1,y1) has the equation y – y1 = m(x – x1) This is called Point-Slope Form Ex. Write the equation in Slope-Intercept Form: a) slope = 3, contains (1,-2) Ex. Write the equation in Slope-Intercept Form: b) contains (2,5) and (4,-1) Ex. Find equation of the lines that pass through (2,-1) are: a) parallel to, and b) perpendicular to, the line 2x – 3y = 5 1 12 Ex. The maximum slope of a wheelchair ramp is . A business is installing a ramp that rises 22 in. over a horizontal distance of 24 ft. Is the ramp steeper than required? Ex. An appliance company determines that the total cost, in dollars, of producing x blenders is C = 25x + 3500 Explain the significance of the slope and y-intercept. Ex. A college purchases exercise equipment worth $12,000. After 8 years, the equipment is determined to have a worth of $2000. Express this relationship as a linear equation. How many years will pass before the equipment is worthless? Practice Problems Section 2.1 Problems 9, 21, 41, 51, 69, 107, 109 Functions Def. (formal) A function f from set A to set B is a relation that assigns to each element x in set A exactly one element y in set B. Def. (informal) A set of ordered pairs is a function if no two points have the same x-coordinate Set A (the x’s) The input Set B (the y’s) The output Domain Range x is called the independent variable y is called the dependent variable Ex. Determine whether the relation is a function: a) x 0 3 2 -1 3 y b) 5 7 2 -2 -3 0 4 Ex. Determine whether the relation is a function: c) x is the number of representatives from a state, y is the number of senators from the same state d) x is the time spent at a parking meter, y is the cost to park Ex. Determine whether the relation is a function: e) x2 + y = 1 f) x + y2 = 1 Rather than writing y = 7x + 2 we can express a function as f (x) = 7x + 2 This is called Function Notation Ex. Let g(x) = -x2 + 4x + 1, find a) g(2) b) g(t) c) g(x + 2) The next example is called a piecewise function because the equation depends on what we are plugging in. 2 x 1x 0 Ex. Let f , find f (-1), x x 1x 0 f (0), and f (1). Finding the domain means determining all possible x’s that can be put into the function Ex. Find the domain of the function f : {(-3,0), (-1,4), (0,2), (2,2), (4,-1)} Often, finding the domain means finding the x’s that can’t be used in the function Ex. Find the domain of the function 1 a) gx x5 r b) Volume of a sphere: V 4 3 3 Ex. Find the domain of the function 2 c) h x 4x Ex. You’re making a can with a height that is 4 times as long as the radius. Express the volume of the can as a function of height. Ex. When a baseball is hit, the height of a baseball is given by the function f (x) = -0.0032x2 + x + 3, where x is distance travelled (in ft) and f (x) is height (in ft). Will the baseball clear a 10-foot fence that is 300 ft from home plate? f x h f x Ex. For f (x) = x2 – 4x + 7, find h Practice Problems Section 2.2 Problems 9, 15, 29, 35, 59, 79, 87, 93