Ch2: Graphs

1. y axis y - $$ in thousands x axis x Yrs Ch2: Graphs Quadrant II (-, +) Quadrant I (+, +) Origin (0, 0) (6,0) -6 -2 4 6 2 y intercept x intercept (5,-2) (-6,-3) (0,-3) Quadrant III (-, -) Quadrant IV (+, -) Graphs represent trends in data. For example: x – number of years in business y – thousands of dollars of profit Equation : y = ½ x – 3 When distinct points are plotted as above the graph is called a scatter plot – ‘points that are scattered about’ A point in the x/y coordinate plane is described by an ordered pairof coordinates (x, y)

2. y x 2.1 Distance & Midpoint • Things to know: • Find distance or midpoint given 2 points • Given midpoint and 1 point, find the other point • Given distance and 1 point, find the other point Origin (0, 0) -6 -2 4 6 2 A point in the x/y coordinate plane is described by an ordered pairof coordinates (x, y) (5,-2) (-6,-3) The Midpoint Formula To find the coordinates of the midpoint (M) of a segment given segment endpoints of (x1, y1) and (x2, y2) x1 + x2, y1 + y2 2 2 The Distance Formula To find the distance between 2 points (x1, y1) and (x2, y2) d =  (x2 – x1)2 + (y2 – y1)2 M

3. y x 2.2 & 2.3 Linear Equations The graph of a linear equation is a line. A linear function is of the form y = ax + b, where a and b are constants. y = 3x + 2 y = 3x + 5x y = -2x –3 y = (2/3)x -1 y = 4 6x + 3y = 12 x y=3x+2 x y=2/3x –1 0 2 0 -1 1 5 3 1 All of these equations are linear. Three of them are graphed above.

4. y x X and Y intercepts Equation: y = ½ x – 3 (6,0) y intercept x intercept (0,-3) -3 The y intercept happens where y is something & x = 0: (0, ____) Let x = 0 and solve for y: y = ½ (0) – 3 = -3 The x intercept happens where x is something & y = 0: (____, 0) Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x => x = 6 6

5. y x Slope Slope is the ratio of RISE (How High) y2 – y1y(Change in y) RUN (How Far) x2 – x1 x(Change in x) = Slope = 5 – 2 = 3 1 - 0 Slope = 1 – (-1) = 2 3 – 0 3 y = mx + b m = slope b = y intercept • Things to know: • Find slope from graph • Find a point using slope • Find slope using 2 points • Understand slope between • 2 points is always the same • on the same line x y=3x+2 x y=2/3x –1 0 2 0 -1 1 5 3 1

6. Zero Slope Positive Slope y y m > 0 m = 0 x x Negative Slope Undefined Slope y y Line rises from left to right. Line is horizontal. m is undefined m < 0 x x Line falls from left to right. Line is vertical. The Possibilities for a Line’s Slope (m) Example: y = ½ x + 2 Example: y = -½ x + 1 Example: y = 2 Example: x = 3 Question: If 2 lines are parallel do you know anything about their slopes? • Things to know: • Identify the type of slope given a graph. • Given a slope, understand what the graph would look like and draw it. • Find the equation of a horizontal or vertical line given a graph. • Graph a horizontal or vertical line given an equation • Estimate the point of the y-intercept or x-intercept from a graph.

7. Linear Equation Forms (2 Vars) Standard Form Ax + By = C A, B, C are real numbers. A & B are not both 0. Example:6x + 3y = 12 • Things to know: • Graph using x/y chart • Know this makes a line graph. Slope Intercept Form y = mx + b m is the slope b is the y intercept Example:y = - ½ x - 2 • Things to know: • Find Slope & y-intercept • Graph using slope & y-intercept Point Slope Form y – y1 = m(x – x1) Example: Write the linear equation through point P(-1, 4) with slope 3 y – y1 = m(x – x1) y – 4 = 3(x - - 1) y – 4 = 3(x + 1) • Things to know: • Change from point slope to/from other forms. • Find the x or y-intercept of any linear equation

8. Parallel and PerpendicularLines & Slopes • Things to know: • Identify parallel/non-parallel lines. PARALLEL • Vertical lines are parallel • Non-vertical lines are parallel if and only if they have the same slope • y = ¾ x + 2 • y = ¾ x -8 Same Slope • Things to know: • Identify (non) perpendicular lines. • Find the equation of a line parallel or • perpendicular to another line through • a point or through a y-intercept. PERPENDICULAR • Any horizontal line and vertical line are perpendicular • If the slopes of 2 lines have a product of –1 and/or • are negative reciprocals of each other then the lines are perpendicular. • y = ¾ x + 2 • y = - 4/3 x - 5 Negative reciprocal slopes 3 • -4 = -12 = -1 4 3 12 Product is -1

9. Practice Problems • Find the slope of a line passing through (-1, 2) and (3, 8) • Graph the line passing through (1, 2) with slope of - ½ • Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ? • Parallel, Perpendicular or Neither? 3y = 9x + 3 and 6y + 2x = 6 • Find the equation of a line parallel to y = 4x + 2 through the point (-1,5) • Find the equation of a line perpendicular to y = - ¾ x –8 through point (2, 7) • Find a line parallel to x = 7; Find a line perpendicular to x = 7 • Find a line parallel to y = 2; Find a line perpendicular to y = 2 • 8. Graph (using an x/y chart – plotting points) and find intercepts of • any equation such as: y = 2x + 5 or y = x2 – 4

10. Symmetry in Graphing Y-Axis Symmetry even functions f (-x) = f (x) For every point (x,y), the point (-x, y) is also on the graph. Test for symmetry: Replace x by –x in equation. Check for equivalent equation. Origin Symmetry odd functions f (-x) = -f (x) For every point (x, y), the point (-x, -y) is also on the graph. Test for symmetry: Replace x by –x , y by –y in equation. Check for equivalent equation. X-Axis Symmetry (For every point (x, y), the point (x, -y) is also on the graph.) Test for symmetry: Replace y by –y in equation. Check for equivalent equation. y = x3 y = x2 Y-axis Symmetry x = y2 X-axis Symmetry Origin Symmetry Symmetry Test -y = (-x)3 -y = -x3 y = x3 Symmetry Test y = (-x)2 y = x2 Symmetry Test x = (-y)2 X = y2

11. A Rational Function Graph & Symmetry • y = 1 • x • x y • -2 -1/2 • -1 -1 • -1/2 -2 • 0 Undefined • ½ 2 • 1 • ½ Intercepts: No intercepts exist If y = 0, there is no solution for x. If x = 0, y is undefined The line x = 0 is called a vertical asymptote. The line y = 0 is called a horizontal asymptote. Symmetry: y = 1/-x => No y-axis symmetry -y = 1/-x => y = 1/x => origin symmetry -y = 1/x => y = -1/x => no x-axis symmetry

12. Application: Solar Energy • The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs, 7.5 feet wide, an equation for the cross-section is: 16y2 = 120x – 225 • (a) Find the intercepts of the graph of the equation • (b) Test for symmetry with respect to the x-axis, y-axis, and origin

13. Application: Cigarette Use A study by the Partnership for a Drug-Free America indicated that, in 1998, 42% of teens in grades 7 through 12 had recently used cigarettes. A similar study in 2005 indicated that 22% of such teens had recently used cigarettes. • Write a linear equation that relates the percent of teens y to the number of years after 1998, x. • Find the intercepts of the graph of your equation. • Do the intercepts have any meaningful interpretation? • Use your equation to predict the percent for the year 2019. Is this result reasonable?

14. 2.5 Proportions & Variation • Proportion equality of 2 ratios. Proportions are used to solve problems in everyday life. • If someone earns $100 per day, then how many dollars can the • person earn in 5 days? • 100 x (x)(1) = (100)(5) • 1 5 x = 500 • 2. If a car goes 210 miles on 10 gallons of gas, the car can go 420 miles on X gallons • 210 420 (210)(x) = (420)(10) • 10 x (210)(x) = 4200 • x = 4200 / 210 = 20 gallons • If a person walks a mile in 16 min., that person can walk a half mile in x min. • 16 x (x)(1) = ½(16) • 1 ½ x = 8 minutes = = =

15. y = kx y is directly proportional to x. y varies directly with x k is the constant of proportionality Example: y = 9x (9 is the constant of proportionality) Let y = Your pay Let x – Number of Hours worked Your pay is directly proportional to the number of hours worked. Direct Variation Example1: Salary (L) varies directly as the number of hours worked (H). Write an equation that expresses this relationship. Salary = k(Hours) L = kH Example 2: Aaron earns $200 after working 15 hours. Find the constant of proportionality using your equation in example1.. 200 = k(15) So, k = 200/15 = 13.33

16. Inverse Variation y = k y is inversely proportional to x x y varies inversely as x Example: y varies inversely with x. If y = 5 when x = 4, find the constant of proportionality (k) 5 = k So, k = 20 4 Example: P. 199 #7 : F varies inversely with d2; F = 10 when d = 5 F = k/ d2 10 = k/52 10 = k/25 K = 250 => F = 250/ d2

17. Another Example (Inverse Variation) y = k y is inversely proportional to x x y varies inversely as x P. 197: Example 2: The maximum weight W that can be safely supported by a 2-inch by 4-inch piece of point varies inversely with its length L. Experimenters indicate that the maximum weight that a 10 foot long 2 x 4 Piece of pine can support is 500 lbs. Write a general formula relating the Maximum weight W that can be safely supported by a length of 25 feet. W = k/L So, 500 = k/10 => k = 5000 Thus, W = 5000/L So, Maximum weight safely supported for Length of 25ft is: W = 5000/25 = 200 lbs.

18. Direct Variation with Power y = kxn y is directly proportional to the nth power of x Example: Distance varies directly as the square of the time (t) Distance = kt2 D = kt2 Joint Variation y = kxp • y varies jointly as x and p

19. Example: (Joint Variation) y = kxp • y varies jointly as x and p P. 198 Example4: The force F of the wind on a flat surface positioned at a right angle To the direction of the wind varies jointly with the area A of the Surface and the square of the speed v of the wind. A wind of 30 miles per hour blowing on a window measuring 4 ft By 5 ft has a force of 150 lbs. What is the force on a window measuring 3 ft by 4 ft caused by a wind of 50 mph? F = kAv2 A = 4 x 5 = 20 v = 30 150 = k (20)(900) 150 = 18000k K = 150/18000 = 1/120 F = (1/120)Av2 , so, when wind = 50mph on 3 x 4 (A = 12), F = (1/120)(12)(2500) = 250lbs

20. Example: Combined Variation P. 199 S#13 The square of T varies directly with the cube of A and Inversely with the square of D T = 2 when A = 2 and D = 4. Write a general formula to describe each variation T2 = kA3 4 = k(8) 8k = 64  k = 8 D2 16 Thus, T2 = 8A3 D2