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Understanding Linear Functions and Relations

Learn about domains, ranges, functions, discrete vs. continuous relations, vertical line test, slope, prediction equations, and graphing inequalities in linear relations and functions.

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Understanding Linear Functions and Relations

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  1. Chapter 2 Linear Relations and Functions

  2. Definitions • DOMAIN: The set of x coordinates from a group of ordered pairs • RANGE: The set of y coordinates from a group of ordered pairs • FUNCTION: a type of relation in which each element of the domain is mapped with EXACTLY one element of the range • ONE-TO-ONE FUNCTION: each element of the range is paired with exactly one element of the domain • DISCRETE: a relation in which the domain is a set of individual points. • CONTINUOUS: a relation with an infinite number of elements and can be graphed continuously as a line or smooth graph. • VERTICAL LINE TEST: used to determine if a relation is a function 2.1

  3. State the Domain and Range. Is the relation a function? • Domain: {-4, -3, 0, 1, 3} • Range: {-2, 0, 1, 2, 3} • It is a function (3,3) (-3,1) (1, 2) (-4,0) (0,-2)

  4. Mapping • {(-1,5) (1,3)(4,5)} -1 1 4 3 5 NOTE this is a function, each x is mapped to exactly one y

  5. You Try Mapping • {(5,6) (-3,0) (1,1) (-3,6)}

  6. Discrete vs. Continuous • Graph y=3x-1 then find the domain and range, determine if it is discrete or continuous

  7. Examples Continued • Graph y=x2 + 1 and find the domain and range. Determine if it is discrete or continuous

  8. Function NotationSubstitute the variable with the term in the parentheses • Given f(x)= x3 – 3 • Find f(1) • Find f(-2) • Find f(2y)

  9. Linear Function f(x)=mx + b *Have a highest exponent of 1 Linear Equation y=mx+b *Have a highest exponent of 1 Standard Form Ax + By = C *A, B, and C must be integers 2.2

  10. Examples • 1. State whether each function is a linear function, explain. • g(x)=2x-5 • g(x) is a linear function because the highest exponent in 1 and it is in slope intercept form m=2 and b = -5 • p(x)=x3+2 • p(x) is not a linear function because x has an exponent > 1 • f(x)= 4+7x • f(x) is a linear function because the highest exponent is 1 and it can be written in slope intercept form with m=7 and b = 4

  11. Example 2 • Write each equation in standard form. Identify A, B, and C

  12. Example 3 • Graph the equation by the intercepts. • Find the x-int and y-int by substituting the other letter with a zero (write as ordered pairs) -2x + y – 4 = 0

  13. Slope: Positive slope Negative Slope Zero Slope Undefined Slope • Parallel Lines have the same slope • Perpendicular lines have slopes that are opposite signs and reciprocals 2.3

  14. Find the slope • A. (1, -3) (3, 5) • B. A line parallel to x – 3y = 3 • C. A line perpendicular to (2, 2) (4, 2)

  15. Graph the line • Passes through (2, -5) parallel to the graph of x = 4 • Passes through the origin perpendicular to the graph of y = -x

  16. Writing Linear Equations • Slope-Intercept Form: y = mx + b • m is slope and b is the y-intercept • Point-Slope Form: y – y1 = m (x – x1) • m is slope and y1 and x1 are any ordered pair on the line 2.4

  17. Write the equation of the line in slope-intercept and point-slope forms • Through (6, 1) and (8, -4) • Through (-5, 7) perpendicular to y = ½x + 6

  18. Scatter Plots 2.5

  19. Steps for Scatter Plots • Graph Ordered Pairs • Select two points to connect for the line of best fit. • Write equation of that line using those two points to find slope • Answer any additional questions using the equation you just wrote.

  20. Find and use a prediction equation EDUCATIONThe table below shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data and draw a line of fit.

  21. Answer Graph the data as ordered pairs, with the number of years since 1985 on the horizontal axis and the percentage on the vertical axis. The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points

  22. Slope formula Substitute. Simplify. Example 1 Cont… Find a prediction equation. What do the slope and y-intercept indicate? • Find an equation of the line through (3, 18) and (15, 13). Begin by finding the slope

  23. Point-slope form Substitute. Distribute. Simplify. Example 1 Cont… Answer: One prediction equation is y = –0.42x + 19.26. The slope indicates that the percent of students sending applications to two colleges is falling about 0.4% each year. The y-intercept indicates that the percent in 1985 should have been about 19%.

  24. Prediction equation x = 25 Simplify. Example 1 Cont… • Predict the percent in 2010 • The year 2010 is 25 years after 1985, so use the prediction equation to find the value of y when x = 25. Answer: The model predicts that the percent in 2010 should be about 9%.

  25. Graphing Inequalities • The equation makes the line to define the boundary • The shaded region is the half-plane • Get the equation into slope-intercept form • Graph the intercept and use the slope to find at least 2 more points • Draw the line (dotted or solid) • Test an ordered-pair not on the line • If it is true shade that side of the line • If it is false shade the other side of the line 2.7

  26. Ex: 3y - 2 > -x + 7 < or > or Dotted Line Solid Line 3y – 2 > -x + 7 +2 +2 3y > -x + 9 /3 /3 /3 y > - x + 3 m = - b = 3 = (0, 3) Test: (0, 0) 0 > - (0) + 3 0 > 0 + 3 0 > 3 false (shade other side)

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