Exit Level TAKS Preparation Unit Objective 3 A
Exit Level TAKS Preparation Unit Objective 3 © A Very Good Teacher 2007
Interpreting Linear Functions can be represented in different ways: y = 2 x + 3 means the same thing as f(x) = 2 x + 3 Linear Functions must have a slope (rate of change) and a y intercept (initial value). In a function… § the slope is the constant (number) next to the variable § the y intercept is the constant (number) by itself 3, Ac 1 A © A Very Good Teacher 2007
Interpreting Linear Functions, cont… • Example: Identify the situation that best 50 represents the amount f(n) = 425 + 50 n. Slope (rate of change) = Y intercept (initial value) = Find an answer that has: 425 as a non-changing value and 50 as a recurring charge every month, every year, etc… Something like Joe has $425 in his savings account and he adds $50 every month. 3, Ac 1 A © A Very Good Teacher 2007
Converting Tables to Equations • When given a table of values, USE STAT! • Example: What equation describes the relationship between the total cost, c, and the number of books, b? b c 10 75 15 100 20 125 25 150 Answer: c = 5 x + 25 3, Ac 1 C © A Very Good Teacher 2007
Converting Graphs to Equations • Make a table of values • Then, use STAT! • Example: Which linear function describes the graph shown below? x y -2 0 5 4 2 3 4 2 Answer: y = -. 5 x + 4 3, Ac 1 C © A Very Good Teacher 2007
Converting Equation to Graph • Graph the function in y = • Example: Which graph best describes the function y = -3. 25 x + 4? Find an answer that has the same y intercept and x intercept as the calculator graph. 3, Ac 1 C © A Very Good Teacher 2007
Equations that are in Standard Form • Sometimes your equations won’t be in y = mx + b form. • They will be in standard form: Ax + By = C • You must convert them to use the calculator! Example: 3 x + 2 y = 12 -3 x 2 y = -3 x + 12 2 3, Ac 1 C Step 1: Move the x Step 2: Divide everything by the number in front of y © A Very Good Teacher 2007
Slope and Rate of Change (m) • Slope and rate of change are the same thing! • They both indicate the steepness of a line. • Three ways to find the slope of a line: By Formula: You must have 2 points on a line By Counting: You must have a graph 3, Ac 2 A By Looking: You must have an equation © A Very Good Teacher 2007
Slope and Rate of Change (m), cont… • By Formula: • Find two points on the graph (they won’t be given to you) (0, 4) and (2, 3) 3, Ac 2 A © A Very Good Teacher 2007
Slope and Rate of Change (m), cont… • By Counting • Find two points on the graph Down 2 Right 4 3, Ac 2 A © A Very Good Teacher 2007
Slope and Rate of Change (m), cont… • By Looking • The equation won’t be in y = mx + b form • You’ll have to change it • If in Standard Form use Process on Slide 7 • If in some other form, you’ll have to work it out… Example: What is the rate of change of the function 4 y = -2(x – 24)? 4 y = -2 x + 24 4 Try to get rid of any parentheses and get the y by itself (isolated). 3, Ac 2 A © A Very Good Teacher 2007
Slope and Rate of Change (m), cont… • Special Cases • Horizontal lines line y = 4 Have slope of zero, m = 0 • Vertical lines like x = 4 Have slope that is undefined 3, Ac 2 A © A Very Good Teacher 2007
m and b in a Linear Function • Changes to m, the slope, of a line effect its steepness y = 1/3 x + 0 • Changes to b, the y y = 1 x + 0 intercept, of a line effect its vertical position (up or down) y = 3 x + 0 y = 1 x + 3 y = 1 x + 0 3, Ac 2 C y = 1 x - 4 © A Very Good Teacher 2007
m and b in a Linear Function, cont… • Parallel Lines have equal slope (m) y = ¼ x – 3 and y = ¼ x + 6 • Perpendicular Lines have opposite reciprocal slope (m) y = ¼ x – 5 and y = -4 x + 15 • Lines with the same y intercept will have the same number for b y=¾x– 9 and y = 5 x – 9 3, Ac 2 C © A Very Good Teacher 2007
Linear Equations from Points • Make a table • USE STAT • Example: Which equation represents the line that passes through the points (3, -1) and (-3, -3)? x 3 -3 y -1 -3 Answer: 3, Ac 2 D © A Very Good Teacher 2007
Intercepts of Lines • To find the intercepts from a graph… just look! • The x intercept is where a line crosses the x axis • The y intercept is where a line crosses the y axis (0, 2) (4, 0) 3, Ac 2 E © A Very Good Teacher 2007
Intercepts of Lines, cont… • To find intercepts from equations, use your calculator to graph them • Example: Find the x and y intercepts of 4 x – 3 y = 12. -4 x -3 y = -4 x + 12 -3 -3 -3 x intercept: (3, 0) y intercept: (0, -4) 3, Ac 2 E © A Very Good Teacher 2007
Direct Variation • Set up a proportion! • Make sure that similar numbers appear in the same location in the proportion • Example: If y varies directly with x and y is 16 when x is 5 what is the value of x when y = 8? 16 x = 5(8) 16 x = 40 16 16 x = 2. 5 3, Ac 2 F © A Very Good Teacher 2007
Direct Variation, cont… • To find the constant of variation use a linear function (y = kx) and find the slope • The slope, m, is the same thing as k • Example: If y varies directly with x and y = 6 when x = 2, what is the constant of variation? The equation for y = kx this situation 3=k 6 = k(2) would be y = 3 x 2 2 3, Ac 2 F © A Very Good Teacher 2007
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