Algebra Unit 2: Linear Functions Notes. Slope Notes. 4 Types of Slope. Slope from a Formula

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1 Undefined Slope Notes Types of Slope Zero Slope Slope can be described in several ways: Steepness of a line Rate of change rate of increase or decrease Rise Run Change (difference) in y over change (difference) in x Slope from a Formula In the above problems with the table, you had to calculate the difference in two y-values first before you calculated the difference in two x-values. This leads us to the slope formula which can be used to calculate the slope of any two points. Slope Formula m = y y 1 x x 1 where (x1, y1) & (x, y) are coordinate points Ex. Calculate the slope of two points using the slope formula. A. (9, 3), (19, -17) B. (1, -19), (-, -7) 1

2 Slope from a Graph Slope can be calculated in several different ways: graphs, tables, formulas, word problems, and equations. Ex. Calculate the slope of each of the graphs. A. Slope: y-intercept: B. Slope: y-intercept: C. Slope: y-intercept: D. Slope: y-intercept: E. Slope: y-intercept: F. Slope: y-intercept:

3 Slope from a Table Calculate the slope using points in the table from our scenario at the beginning of the lesson. (Remember slope is the change in y divided the change in x.) a. b. Real World Slopes If a graph, table, equation, or context represents a real world situation, the slope has a meaning that can be interpreted as a rate of change. For the following representations, calculate the slope and interpret it as a rate of change. a. b. Slope/Rate of Change: Slope/Rate of Change: Unit Rate of Change: Unit Rate of Change: 3

4 Slopes & Y-intercepts Notes A y-intercept is the point where the graph crosses the y-axis. Its coordinate will always be the point (0, b), where b stands for the number on the y-axis where the graph crosses and the value of the x-coordinate will always be 0. Ex. Identify the y-intercept in the following representations: A. B. C. D. Real World Y-Intercepts In a real world situation, the y-intercept represents the starting value or starting point. Determine the y-intercept for the following table: A. How many pills were in the bottle to start?

5 Graphing Linear Equations When we write an equation of a line, we use slope intercept form which is y = mx + b, where m represents the slope and b represents the y-intercept. Slope Intercept Form y = mx + b m: slope b: y=intercept Ex. Going back to yesterday s notes, since you know the slope and y-intercept, create the equation for each line. Slope and Y-intercepts from an Equation The equation for a line includes and represents the slope and y-intercept. The equation for a line is y = mx + b, where m is the slope and b is the y-intercept. It is called slope intercept form. Slope Intercept Form y = mx + b m: slope b: y-intercept a. y = -x + 1 b. 3x y = -1 Slope: y-intercept: Slope: y-intercept: 5

6 Graphing Linear Functions When you graph equations, you have to be able to identify the slope and y-intercept from the equation. Step 1: Solve for y (if necessary) Step. Plot the y-intercept. Step 3: From the y-intercept, use the slope to calculate another point on the graph. Step : Connect the points with a ruler or straightedge. Slope = change in y = + change in x + Ex. Graph the following lines: A. y = x + m = b = y = 3x + m = b = C. y = -x 1 m = b = D. y = 5 x 3 m = b =

7 Graphing Horizontal and Vertical Lines When graphing horizontal and vertical lines, you will have one variable set equal to a constant. Whatever constant the variable is set equal to represents that value in a coordinate point. For example, if you have y =, all coordinate points must have a value of and x can be whatever you want. Pick 3 points to graph the lines below. Ex. y = Ex. x = Ex. x = 3 Ex. y =

8 Graphing Linear Inequalities A linear inequality is similar to an equation as you learned before, but the equal sign is replaced with an inequality symbol. A solution to an inequality is any ordered pair that makes the inequality true. Ex. Tell whether the ordered pair is a solution to the inequality. (7, 3); y < x 3 (, 5); y < x + 1 (, 5); y x + 1 A linear inequality describes a region of a coordinate plane called a half-plane. All the points in the shaded region are solutions of the linear inequality. The boundary line is the line of the equation you graph. Symbol Type of Line Shading < Dashed Below boundary line > Dashed Above boundary line Solid Below boundary line Solid Above boundary line Graphing Linear Inequalities Step 1: Solve the inequality for y (if necessary). Step : Graph the boundary line using a solid line for or OR a dashed line for < or >. Step 3: If the inequality is > or, shade above the boundary line If the inequality is < or, shade below the boundary line OR Select a test point and substitute it into linear inequality. If the test point gives you a true inequality, you shade the region where the test point is located. If the test point gives you a false inequality, you shade the region where the test point is NOT located.

9 Practice Graphing Linear Inequalities Ex. Graph the inequality: Ex. Graph the inequality: a. y < 3x + b. y 3 x Test Point: Test Point: Ex. Graph the inequality: Ex. Graph the inequality: a. 3x + y b. x 3y > Test Point: Test Point: 9

10 Naming Linear Inequalities Practice: Name each linear inequality from the graph: a. b. c. d. 10