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Understanding Linear Functions: Representation, Slope, and Intercepts

This guide covers the essential concepts of interpreting linear functions, including their various forms like equation, graph, and table. It explains how to identify the slope (rate of change) and the y-intercept (initial value) in different representations. Additionally, it illustrates how to convert between tables, equations, and graphs, and discusses the relationships between slope, y-intercept, and characteristics of lines such as parallelism and perpendicularity. Key principles of direct variation are also summarized to aid comprehension.

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Understanding Linear Functions: Representation, Slope, and Intercepts

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  1. Exit Level TAKS Preparation Unit Objective 3

  2. Interpreting Linear Functions Functions can be represented in different ways: y = 2x + 3 means the same thing as f(x) = 2x + 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, A.05A

  3. Interpreting Linear Functions, cont… 425 50 • Example: Identify the situation that best represents the amount f(n) = 425 + 50n. 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, A.05A

  4. 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? Answer: c = 5x + 25 3, A.05C

  5. Converting Graphs to Equations • Make a table of values • Then, use STAT! • Example:Which linear function describes the graph shown below? -2 5 0 4 2 3 4 2 Answer: y = -.5x + 4 3, A.05C

  6. Converting Equation to Graph • Graph the function in y = • Example: Which graph best describes the function y = -3.25x + 4? Find an answer that has the same y intercept and x intercept as the calculator graph. 3, A.05C

  7. 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: 3x + 2y = 12 Step 1:Move the x -3x -3x 2y = -3x + 12 Step 2:Divide everything by the number in front of y 2 2 2 3, A.05C

  8. 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: By Counting: By Looking: You must have 2 points on a line You must have a graph You must have an equation 3, A.06A

  9. 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, A.06A

  10. Slope and Rate of Change (m), cont… • By Counting • Find two points on the graph Down 2 Right 4 3, A.06A

  11. 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 4y = -2(x – 24)? Try to get rid of any parentheses and get the y by itself (isolated). 4y = -2x + 48 4 4 4 3, A.06A

  12. 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, A.06A

  13. m and b in a Linear Function • Changes to m, the slope, of a line effect its steepness y = 1/3x + 0 y = 1x + 0 • Changes to b, the y intercept, of a line effect its vertical position (up or down) y = 3x + 0 y = 1x + 3 y = 1x + 0 y = 1x - 4 3, A.06C

  14. 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 = -4x + 15 • Lines with the same y intercept will have the same number for b y = ¾ x – 9 and y = 5x – 9 3, A.06C

  15. 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)? Answer: 3, A.06D

  16. 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, A.06E

  17. Intercepts of Lines, cont… • To find intercepts from equations, use your calculator to graph them • Example:Find the x and y intercepts of 4x – 3y = 12. -4x -4x -3y = -4x + 12 -3 -3 -3 x intercept: (3, 0) y intercept: (0, -4) 3, A.06E

  18. 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? 16x = 5(8) 16x = 40 16 16 x = 2.5 3, A.06F

  19. 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 this situation would be y = 3x y = kx 3 = k 6 = k(2) 2 2 3, A.06F

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