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Chapter 8 Review Section 8.1 “Relations and Functions” RELATION a pairing of numbers from one set, called the DOMAIN, with the numbers in another set, called the RANGE. Set 1 Set 2 Domain Input Range Output X-coordinate Y-coordinate FUNCTIONIs a relation in which for each input there is EXACTLY ONE output. FUNCTION 2 4 6 8 0 1 0 1 NOT A FUNCTION 3 5 5 1 2 3 Each input must be paired with only ONE output Solving Linear Equations To find a SOLUTION, substitute the ordered pair into the equation and see if it produces a true statement. 2x – y = 5 Tell whether the ordered pair is a solution to the equation. point NO YES (1, 3) (2,-1) equation substitution 2x - y = 5 2(1) – 3 = 5 2x - y = 5 2(2) – -1 = 5 check -1 = 5 5=5 Linear Equationan equation whose graph is a line 4x + y = 9 Function formAn equation is in function form when the equation is solved for y. 4x + y = 9 y = 9 – 4x Solve the equation for y. Section 8.3 “Using Intercepts” x-intercept- y-intercept- the x-coordinate of the point where the graph crosses the x-axis the y-coordinate of the point where the graph crosses the y-axis To find the x-intercept, solve for ‘x’ when ‘y = 0.’ Find the x-intercept of the graph 2x + 7y = 28. 2x + 7(0)= 28 2x = 28 x = 14 To find the y-intercept, solve for ‘y’ when ‘x = 0.’ Find the y-intercept of the graph 2x + 7y = 28. 2(0)+ 7y= 28 7y = 28 y=4 Section 8.4 “The Slope of a Line” SLOPEthe ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on a line. Slope = rise = change in y run change in x Slope Review The slope m of a line passing through two points ( x1 , y1 ) and ( x2 , y2 ) is the ratio of the rise change to the run. y m ( y2 y1 ) ( x2 x1 ) run ( x2 , y2 ) rise ( x1 , y1 ) x Section 8.5 “Graph Using Slope-Intercept Form” SLOPE-INTERCEPT FORMa linear equation written in the form y-coordinate x-coordinate y = mx + b slope y-intercept Parallel Lines two lines in the same plane are parallel if they never intersect. Because slope gives the rate at which a line rises or falls, two lines with the SAME SLOPE are PARALLEL. y = 3x + 2 y = 3x – 4 Perpendicular Lines two lines in the same plane are perpendicular if they intersect at right angles. Because slope gives the rate at which a line rises or falls, two lines with slopes that are NEGATIVE RECIPROCALS are PERPENDICULAR. ½ and -2 are negative reciprocals. 4/3 and -3/4 are negative reciprocals. y = -2x + 2 y = 1/2x – 4 Section 8.6 “Writing Linear Equations” You can write a linear equation in slopeintercept form, if you know the slope (m) and the y-intercept (b) of the equation’s graph. SLOPE-INTERCEPT FORMa linear equation written in the form y-coordinate slope x-coordinate y = mx + b y-intercept Write an equation of the line that passes through the given points. (-6, 0), (0, -24) STEP 1 Calculate the slope of the line that passes through (-6, 0) and (0, -24). STEP 2 Write an equation of the line. The line crosses the y-axis at (0, -24). So, the y-intercept is -24. y = mx + b Write slope-intercept form. y = -4x + (-24) Substitute -4 for m and -24 for b. The equation is y = -4x – 24. Section 8.7 “Function Notation” Function Notationa linear function written in the form y = mx + b where y is written as a function f. x-coordinate This is read as ‘f of x’ f(x) = mx + b slope f(x) is another name for y. It means “the value of f at x.” g(x) or h(x) can also be used to name functions y-intercept Linear Functions What is the value of the function f(x) = 3x – 15 when x = -3? A. -24 B. -6 C. -2 f(-3) = 3(-3) – 15 Simplify f(-3) = -9 – 15 f(-3) = -24 D. 8 Linear Functions For the function f(x) = 2x – 10, find the value of x so that f(x) = 6. f(x) = 2x – 10 Substitute into the function 6 = 2x – 10 Solve for x. 8 = x When x = 6, f(x) = 8