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# Graphing Skills

 Algebra The Easy Way – Chapter 9 Graphing Skills Objectives • Distance between two points • Graphing solutions to equations • Calculating the slope of a line • Graphing inequalities and curves • Graphing relationships Graphing Data, and Finding Distances • Good overview at the start of chapter - based on compass direction – Two dimensions: North -> up, East -> to the right – Origin at 0 East, 0 North (bottom left corner) • Label a point in space using an ordered pair (x, y), with x representing the East distance, and y the North – By convention, x always stated first – The x and y in a point (x, y) are called a coordinates • Important task: find distance from origin to a point, or between two points – General formulas: from origin: – Between two points: Graphing Equation Solutions • Observation: nice connection between functions (example: y=f(x)) and graphs, if we change – North -> y, East -> x (call these “x and y axes”) – Origin at (0,0) • Solutions to a function like y=2x can be plotted as points – (1,2), (2,4), (3,6), (4,8), etc. (Fig. 9-11) – Set of points representing solutions: graph of equation – Negative numbers legal too: extend axes (Fig. 9-16) • Figs. 9-17 - 9-19 show a number of other examples – All go through the origin, and represent y = f(x) – General form of linear function: y = f(x) = ax + b • Two points required to uniquely determine a line The Slope of a Line • Slope: angle of a graph relative to horizontal – Can be positive, zero, or negative (Fig. 9-26) – Calculating slope (Fig. 9-27): • Pick two points (x1,y1), (x2,y2) • Calculate distance up (along y axis) • Divide by distance sideways (along x axis) • General formula (works for negative slopes): – For equations of the form y = mx + b, all equations with the same “m” have the same slope • If m < 0, slope is negative • The “b” represents the y-intercept: point at which the graph crosses the y axis • Set y=0 to find x-intercept, x=0 to find y-intercept (p. 126) Graphing Inequalities • Often useful to graph inequalities, e.g. x + y ≤ 10 – Assume x, y ≥ 0 – Fig. 9-30 shows integer values that make the equation true (sum of x and y is ≤ 10) – If we draw a line along the boundary, points below make the equation true, points above make it false • More on this subject later • So for an equation ax + by ≤ c (or ≥ c) we draw the line ax + by = c – All points on one side make the inequality true; all points on the other make it false Graphing Curves, other Functions • Many equations cannot be drawn using a straight line – Example: y = x2(see Fig. 9-34, 9-35, 9-36) – Also called a nonlinear (“not a line”) equation – Note the effect shown in Fig. 9-35 of a multiplier--changes the shape (steepness) of the curve • Functions may also be discontinuous: points on the line are not joined by a smooth curve (Fig. 9-37) – Simple continuity test: function can be drawn without lifting your pencil from the paper • Most business data can be massaged into graphical format, and the old adage of a picture being worth a thousand words will often hold true Displaying Graphical Relationships • Book discusses several examples of using graphs to highlight correlations between data elements – Temperature and gas volume – Rainfall and visits to a park • Note negative slope in Fig. 9-39: visits decrease as rainfall increases – Important to distinguish independent (x) variable from dependent (y) variable • Example in Fig. 9-40 shows an incorrect comparison – Hamburgers and library books (?) – Turns out another variable (population) is the real independent variable • Be careful to avoid this problem Algebra The Easy Way – Chapter 10 Systems of Two Equations Objectives• Goal: find solution to a set of two equations • Graphical representation of solution • Simultaneous equation solution methods – Substitution method – Elimination method • General formulas for solutions Catching the Culprit • Problem from book: catching the “culprit” – Detective starts at position 2, moves at speed 4 • y = f(x) = 2 + 4x – Culprit starts at position 10, moves at speed 3 • y = g(x) = 10 + 3x – Both start at the same instant – When will their paths cross? • Initial insight: the two functions can be graphed simultaneously (Fig. 10-4) – Crossing point shows when the culprit will be caught – Looks like at position 34 (right at the river) – How long will the chase take (what is value of x)? Solving Simultaneous Equations • Since both equations are a function of x, we can set them equal to each other, then solve for x: 2 + 4x = 10 + 3x 2 + 4x - 3x = 10 + 3x - 3x (Subtract 3x from both sides) 2 + x = 10 2 - 2 + x = 10 - 2 (Subtract 2 from both sides) x = 8 • Now plug solution for x back into either equation: y = f(x) = 2 + 4(8) y = 34 • So graphical solution was correct Another Example: Supply & Demand • Quantity supplied, demanded is a function of price p – Supply: q = f(p) = -5 + 30p • Suppliers will supply more as price increases (positive slope) – Demand: q = g(p) = 200 - 15p • Consumers will demand less as price increases (negative slope) • Solving for p, we get p = 4.556, q = 131.7 • Graph of solution in Fig. 10-5 • Key assumption in this “economy”: other things equal – More on this later… Equation Substitution • Solve for one variable or the other, then substitute into the other equation – If we have – Solving for x in the first equation we get – Then plug that value for x into the second: – We’ll do the example at the bottom of P. 151 • General process (algorithm): – Use one equation to find y in terms of x (or x in terms of y) • Experience determines which to substitute for the other – Substitute for y in the other equation – Solve for x (or y) – Insert result into first equation to determine value of y (or x) The Elimination Method • For a pair of equations with two unknowns: 4x + 3y = 38 6x - 3y = 12 – We can add the two equations together (4x + 6x) + (3y - 3y) = 38 + 12 – Conveniently, the y term “drops out”: we can solve for x – No violation of “golden rule”: two sides of second equation are equal, so we are adding the “same thing” to both sides • Tricky part: getting the coefficients to match so one drops out • Can also subtract one equation from the other if all coefficients are positive The Elimination Method: Algorithm • Two equations, with two unknowns (x, y) 1. Make coefficients of x or y match, usually by multiplying. It is legal to multiply the two equations by different numbers 2. Subtract the two equations from each other (left-hand side from left-hand side, right-hand side from right-hand side) 3. Result will be a new equation with one variable eliminated: solve for the other variable 4. Plug the result into either of the original equations and solve for the other variable • In step 1, may also set things up so coefficients are negatives of each other--then add in step 2 Notes on Systems of Two Equations • General formulas for elimination, substitution methods are given on p. 156 – Not useful for real-world problems, but nice to know they exist • Both methods are useful, and only experience will help decide which is better for a given problem • Equation system will have no solutions when the two equations are parallel lines • Equation system will have an infinite number of solutions of the two equations define the same line (example p. 158) – Not always obvious this is the case