Math 95, Mod 1, Sec. 3.3 Graphing Linear functions A. Graphing Linear functions E. 1: Graph: f() = 3, g() = 3 + 2 and h() = 3-3 f() g() h() 0-2 2 What happens? B. Linear equations can be written in the form y = m+b or f() = m+b which is called: Or A + By = C which is called: E. 2: Which of the following equations are linear? a) y 3 4 b) y 2 c) 4 2y 8 d ) y 2 e) y 6 1 f ) y 2 3
C. Graphs: E. 3: Which of the following graphs represent linear functions (caution!)? D. Finding - and y-intercepts To find an -intercept, let f()= 0 and solve for. To find a y-intercept, let = 0 and solve for f(). E. 4: Graph each linear function by finding - and y-intercepts. Then write each equation using functional notation. a) y = -4 b) 2y = 8
E. Graphing Vertical and Horizontal Lines: A vertical line is of the form, where c is a real number, with -intercept (, ) A horizontal line is of the form, where c is a real number, with y-intercept (, ) y E. 5: Application The cost of renting a piece of machinery is given by the linear function C() = 4 + 10, where C9) is in dollars and is given in hours. a) Find the cost of renting the machine for 8 hours. b) Graph the cost function. c) How can you tell from the graph of C() that as the number of hours increases, the total cost increases also?
E. 6: Application While manufacturing two different camera models, Kodak found that the basic model costs $55 to produce, where as the delue model costs $75. The weekly budget for these two models is limited to $33,000 in production costs. The linear equation that models this situation is 55 + 75y = 33,000, where represents the number of basic models and y the number of delue models. a) Complete the ordered pair solution (0, ) of this equation. Describe the meaning. b) Complete the ordered pair solution (, 0 ) of this equation. Describe the meaning. c) If 350 delue models are produced, find the greatest number of basic models that can be made in a week.
Slope of a Line: I. Definition: m = E. 7: Find the slope of a line that goes through (-1, -2) and (2, 5). E. 8: Find the slope of a line that goes through (1, -1) and (-2, 4). E. 9: Find the slope of a line that goes through (1, 3) and (4, 3). E. 10: Find the slope of a line that goes through (2, -4) and (-2, -4). E. 11: Review slope: a) Positive slope: b) Negative slope: c) Zero slope: d) Undefined slope:
II. Linear Equations in slope-intercept form: E. 12: Write an equation in slope-intercept form for the line with slope of 1/2 and going through the point (0,3). IV. Parallel Lines: Two nonvertical lines are parallel if they have the but different. E. 14: On the same set of aes, graph: y = 2 + 2 y = 2-1 y V. Perpendicular Lines: Perpendicular lines intersect at a. Two nonvertical lines are perpendicular if the product of their slopes is. E. 15: On the same set of aes, graph: y 2 3 1 y y 2 1 1 2
E. 16: State whether each pair of equations represent parallel lines, perpendicular lines or neither. a) 2 + 5y = 1 4 + 10 y = 3 b) - 4y = 3 3 + 12y = 7 c) -2 + 3y = 1 3 + 2y = 12
VI. Applications: E. 17: As a gas, like helium, is heated, it will epand. The formula: V(t) = 0.147t + 40 calculates the volume in cubic inches of a sample of helium at the temperature, t, in degrees Centigrade (Celsius). a) Is this a linear function? How do you know? b) Evaluate V(0) and interpret what it means. c) What does the number 0.147 mean in terms of this situation? d) If the temperature increases by 10 degrees, how much does the volume of this sample increase? e) What is the volume of this sample of helium at 100 degrees? f) At what temperature will this sample of helium reach a volume of 50 cubic inches?