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Chapter 3 Class Notes Intermediate Algebra, MAT1033C SI Leader Joe Brownlee Palm Beach State College Class Notes 3.1 Professor Burkett SI Leader Joe Brownlee 3.1 β The Rectangular Coordinate System An ordered pair, a βpointβ: Plotted points on a graph: 1 Class Notes 3.1 Professor Burkett SI Leader Joe Brownlee A linear equation in standard form is written π΄π₯ + π΅π¦ = πΆ, where A, B, and C are real numbers, and A and B are not both 0. Example 1: Find ordered pairs that satisfy the given equation. 3π₯ β 4π¦ = 12 x 0 4 y -3 0 Step 1: Begin by substituting 0 for π₯ in the given equation, then solve for π¦. 3(0)-4y=12 -4y=12 y=-3 Step 2: Now substitute 0 for π¦ in the given equation, then solve for π₯. 3x-4(0)=12 3x=12 x=4 We now have two points, (0, -3) and (4, 0), that we can plot on a graph to draw our line. 2 Class Notes 3.1 Professor Burkett SI Leader Joe Brownlee Example 2: Find ordered pairs that satisfy the given equation. 2π₯ β 5π¦ = 10 x 0 5 y -2 0 Step 1: Begin by substituting 0 for π₯ in the given equation, then solve for π¦. 2(0)-5y=10 -5y=10 y=-2 Step 2: Now substitute 0 for π¦ in the given equation, then solve for π₯. 2x-5(0)=10 2x=10 x=5 We now have two points, (0, -2) and (5, 0), that we can plot on a graph to draw our line. Two useful points for graphing are the x- and y-intercepts. The x-intercept is the point, if any, where the line intersects the x-axis. The y-intercept is the point, if any, where the line intersects the y-axis. The above two examples show how to find the x- and y-intercepts of a line. When graphing the equation of a line, find the intercepts as follows: Let π¦ = 0 to find the x-intercept; let π₯ = 0 to find the y-intercept. 3 Class Notes 3.1 Professor Burkett SI Leader Joe Brownlee When graphing horizontal and vertical lines, the equation of the line will indicate which axis will be intersected: Example 3: Graph π¦ = 3. Example 4: Graph π₯ = 1. 4 Class Notes 3.1 Professor Burkett SI Leader Joe Brownlee When finding the midpoint between two endpoints, use the following: Example 5: Find the coordinates of the midpoint of the line segment ππ with endpoints π(4, β3) and π(6, β1). Step 1: Substitute the given coordinates into the midpoint formula, then simplify if possible. ( 4+6 β3+(β1) 2 , 2 10 β4 (5, β2) )= 2 )=( 2 , Step 2: Therefore, the midpoint of line segment ππ is (5, β2). 5 Class Notes 3.2 Professor Burkett SI Leader Joe Brownlee 3.2 β The Slope of a Line The slope of a line simply describes a lineβs steepness, also known as pitch or grade. This sign, used in real life, indicates a steep slope so truck drivers know to slow down when going downhill. To find the slope π of the line through the distinct points (π₯1 , π¦1 ) and (π₯2 , π¦2 ) use the formula: 1 Class Notes 3.2 Professor Burkett SI Leader Joe Brownlee Example 1: To find the slope of the line shown on the graph on the previous page with points (1,1) and (5,-2), simply plug the points into the slope formula and solve: Step 1: Plug points into slope formula. β2β1 β3 β3 ( 5β1 )= ( 4 )= 4 Step 2: So the slope of the line shown on the graph is β3 4 . The slope of a horizontal line is 0. The slope of a vertical line is undefined. To find the slope of a line when given an equation, we must put the equation into slope-intercept form, which is π¦ = ππ₯ + π, where π is the slope and π is the y-intercept (0,b). Example 2: Given the equation 3π₯ β 4π¦ = 12, find the slope of the line and graph. Step 1: Take the given equation, which is written is standard form, and put it into the form y=mx+b 3x-4y=12 -4y=-3x+12 3 π¦ = 4π₯ β 3 3 Step 2: Therefore, 4 is the slope of the line. 2 Class Notes 3.2 Professor Burkett SI Leader Joe Brownlee Step 3: Now that our line is in y=mx+b form, we can quickly and easily graph our line. Since b in this equation is -3, we know this line crosses the y-intercept at (0,-3). Go to that point on a graph and draw a dot. From 3 that dot, since our slope is 4, we will go up 3 units and to the right 4 units. The slopes of parallel lines are the exact same. The slopes of perpendicular lines are opposite reciprocals. 3.2 review at a glance: 3 Class Notes 3.3 Professor Burkett SI Leader Joe Brownlee 3.3 β Linear Equations in Two Variables When given the slope and y-intercept of a line and asked to write the corresponding equation, use the form y=mx+b. Example 1: Write an equation of the line with slope -4 and y-intercept (0,8). π¦ = β4π₯ + 8 Point-Slope Form The point-slope form of the equation of a line with slope π passing through the point (ππ , ππ ) is: ; m=25 1 Class Notes 3.3 Professor Burkett SI Leader Joe Brownlee Example 2: Therefore, the equation for the line passing through the above given points (3, -1) with a slope of 2 is: π¦ = 2π₯ β 7. Parallel and Perpendicular Lines When given two equations of lines and asked whether they are parallel or perpendicular, you must first put both lines in y=mx+b and then compare their slopes. Notice that since the slopes, m, are the same, they are parallel. 2 Class Notes 3.3 Professor Burkett SI Leader Joe Brownlee Notice that since the slopes are opposite (one is negative, the other positive) reciprocals (the numerator and denominator have flipped), the lines are now perpendicular. 3 Class Notes 3.4 Professor Burkett SI Leader Joe Brownlee 3.4 β Linear Equations in Two Variables The solution of a linear inequality in two variables like Ax + By > C is an ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality. Example 1: Is (1, 2) a solution to the given inequality? The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. Graphing a Linear Inequality Step 1: Draw the graph of the straight line that is the boundary. The boundary line is dashed for > and < and solid for β€ and β₯. The half-plane that is a solution to the inequality is usually shaded. Step 2: Choose a test point. Choose any point not on the line, and substitute the coordinates of this point in the inequality. Step 3: Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequality. Otherwise, shade the region on the other side of the boundary line. Example 2: Graph the inequality Since the graph is shaded above the line, we know any point we choose to substitute into the original inequality from that shaded region will make the inequality true. 1 Class Notes 3.4 Professor Burkett SI Leader Joe Brownlee Graphing the Intersection of Two Inequalities Sometimes you will be asked to graph two inequalities on the same grid. ο· If the inequalities you are graphing are separated by βANDβ your answer will be only the section of the graph that is doubled shaded. ο· If the inequalities you are graphing are separated by βORβ your answer will be any part of the graph that is shaded. Example of an AND graph (just the portion of the graph that is doubled shaded is your answer): Example of an OR graph (all shaded areas are your answer): 2 Class Notes 3.5 3.5 β Intro to Relations and Functions Relation: an ordered pair: (x,y) Function: a relation in which for every x, there is only one y. Example 1: x 0 -1 -1 y 5 4 3 This is NOT a function, because x has two possible y-values, 3 or 4. Example 2: x 0 -1 -3 y 1 3 5 This IS a function, because for every x, there is only one y. Domain: the set of all x-values Range: the set of all y-values Example 3: (4, 0), (4, 1), (4, 2) The domain is {4} The range is {0, 1, 2} You can determine if a graph is a function by using a simple test called the vertical line test. If every vertical line you draw on a graph intersects the relation (line) in no more than one point, then the relation is a function. None of the graphs to the right are functions, because they fail our vertical line test. 1 Professor Burkett SI Leader Joe Brownlee Class Notes 3.5 You can determine the domain and range of a graph, too: 2 Professor Burkett SI Leader Joe Brownlee Class Notes 3.6 3.6 β Function Notation and Linear Functions In function notation, π¦ = π(π₯) π(π₯) is pronounced βf of xβ or βf at x.β Example 1: π¦ = 9π₯ β 5 is the same as π (π₯ ) = 9π₯ β 5 Example 2: Let π(π₯ ) = 6π₯ β 2. Evaluate the function π for each of the following: (a) π₯ = β2 π (π₯ ) = 6π₯ β 2 π (β2) = 6(β2) β 2 π (β2) = β12 β 2 π (β2) = β14 Therefore, (-2, -14) is the ordered pair that belongs to π. Example 3: Let π(π₯ ) = 5π₯ β 1. Find and simplify π(π + 2). π(π₯ ) = 5π₯ β 1 π(π + 2) = 5(π + 2) β 1 π(π + 2) = 5π + 10 β 1 π(π + 2) = 5π + 9 1 Professor Burkett SI Leader Joe Brownlee Class Notes 3.6 Professor Burkett SI Leader Joe Brownlee Example 4: For each function, find π(β2). (a) π = { (0, 5), (β1, 3), (β2, 1) } Look for the ordered pair that has a -2 for the x-coordinate. The corresponding y-coordinate will be your answer. Therefore, π(β2) = 1 When youβre asked to graph a function, all you do is treat the function notation ( "π(π₯)" or "π(π₯)" ) as a βyβ and plot the line as usual. Example 5: Graph the function and give the domain and range. 3 π (π₯ ) = 4 π₯ β 2 π¦= 3 π₯β2 4 D= (ββ, β) R= (ββ, β) 2