Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
BKL singularity wikipedia , lookup
Debye–Hückel equation wikipedia , lookup
Equations of motion wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Equation of state wikipedia , lookup
Differential equation wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Calculus of variations wikipedia , lookup
Exact solutions in general relativity wikipedia , lookup
Name:__________________________ Algebra 2 PAP Linear Parent Function Average Rate of Change (Slope) Graphically: Rise Run m= Numerically: y f ( x2 ) f ( x1 ) x x2 x1 Analytically: f ( x) mx b, m is slope Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes. Horizontal lines have a slope of zero. Vertical lines have an undefined slope. Practice: 1. Determine the average rate of change between the two linear points (-1, 6) and (-4, 7). 2. Determine the average rate of change for f(3) = 7 and f(5) = 11. 3. Find the average velocity for y x f ( x) x 2 2 x over the interval [-3, -1]. 4. What is the slope of the line perpendicular to the line x = 2? y x 5. Determine the average rate of change for 2 x 3 y 17 0 . 1. On the same coordinate grid, graph the following equations: y 2 x and y 2 . Graphing Linear Equations Slope-Intercept Form y = mx + b 1) Make sure the equation is solved for y. 2) m is slope and b is y-intercept (where the point crosses the y-axis) 3) To graph, plot the y-intercept first. At the y-intercept, use the slope to count rise over run. Standard Form Ax + By = C 1) A cannot be a negative number or a fraction. 2) B cannot be a fraction. 3) To graph, use the x-intercept and y-intercept. 4) Identify the x-intercept by letting y = 0 and solving for x. 5) Identify the y-intercept by letting x = 0 and solving for y. Point-Slope Form y – y1 = m(x – x1) 1) m is slope and ( x1 , y1 ) is a given coordinate 2) To graph, plot the given point. At the point, use the slope to count rise over run. Practice: 1. Graph the equation y 2 x 1 3 2. Graph the equation 5 x 2 y 10 1 3 3. Graph the equation ( y 5) ( x 2) Additional Practice: Graph each linear equation below on your own graph paper. 1 1. y x 3 3 2. y 4 x 2 3. 6 x 3 y 18 0 4. 2 x 3 y 6 (what are the x and y intercepts?) (what are the x and y intercepts?) 5. y 5 1 ( x 3) 2 6. y 2 2( x 4) Writing the Equation of a Line Write the equation of a line when given… 1. Slope and y-intercept a. Substitute m and b into y = mx + b 1. m = 4 and b = -10 2. Perpendicular to y – 1 = -3/4 (x + 2) 2. A Graph a. Find the y intercept from the graph b. Count the slope from the graph c. Substitute the slope and y intercept in the slope intercept form of the equation. 3. Slope and one point a. Use the point slope formula b. Substitute m and the given point into the formula 1. m = -2 and (-1/2,5) 2. || to 3x – 5y = 4 through (-2,10) 4. Two points a. Calculate the slope of the two points b. Use one of the points and the slope to substitute into the point slope formula. 1. (-2,5) and (3,-1) 2. (1/2,3/4) and (6,-2/7) 5. x and y intercepts a. Write the intercepts as ordered pairs b. Calculate the slope c. Substitute the slope and the y intercept into the slope intercept formula. 1. x-int (-2,0) y-int (0,1/2) Solve Systems of Equations by Graphing Possibilities for graphs of systems of linear equations: Case 1: one solution Case 2: infinite solutions Case 3: no solution When solving by graphing, convert both equations to slope-intercept form OR use standard form graphing with x and y intercepts then graph to find solution. Example 1 : Solve by graphing 4x y 8 2 x 3 y 18 Solution __________________ Example 2 : Solve by graphing Example 3: 2x y 4 2x y 1 y = x -2 y = x+2 y=2 y=x Identify the domain and range of each relation in interval notation. State whether or not each relation is a function. 7. 8. 10. 11. 9. Solving Systems of Equations by Graphing Solve the linear systems by graphing. Write your answer as an ordered pair (x, y). 1. x + 2y = 2 x – 4y = 14 2. y = 3x + 2 y = 3x - 2 5. y = -3x - 13 -x – 2y = -4 3. x – 7y = 6 -3x + 21y = -18 4. y = x -1 y = -x + 4 6. y = x +5 y = -5 Extra Graphs: