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MID-TERM REVIEW NOTES

MID-TERM REVIEW NOTES. DO NOT LOSE THESE!! WE WILL ADD TO THESE DAILY. REAL NUMBERS. Real numbers are every number that can be found on a number line. NOT A REAL NUMBER (FAKE). Any expression that has zero as the denominator. REAL NUMBERS INCLUDE:.

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MID-TERM REVIEW NOTES

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  1. MID-TERM REVIEW NOTES DO NOT LOSE THESE!! WE WILL ADD TO THESE DAILY

  2. REAL NUMBERS • Real numbers are every number that can be found on a number line. NOT A REAL NUMBER (FAKE) • Any expression that has zero as the denominator.

  3. REAL NUMBERS INCLUDE: • RATIONAL NUMBERS- any number that can be written as a fraction {integers and fractions} • INTEGERS- whole numbers (counting numbers including 0) AND their opposites (negatives) {…-3,-2,-1,0,1,2,3…} • WHOLE NUMBERS- counting numbers including zerO, {0,1,2,3…} • NATURAL NUMBERS- counting numbers, {1,2,3…} • IRRATIONAL NUMBERS- any number that cannot be written as a fraction {square root of a non-perfect square and pi}

  4. REAL NUMBERS Ray Found Me Packing Ice In The Restaurant • Ray – rational numbers ARE… • Found – fractions • Me – mixed numbers • Packing – percents % • Ice – integers • In – improper fractions • The – terminating decimals • Restaurant – repeating decimals

  5. SYSTEMS OF EQUATIONS • A system of equations is when you have two or more equations using the same variables. • The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair, no solution, or infinitely many solutions. • Three different methods: Graphing, Substitution, Elimination

  6. Review: Graphing with slope-intercept • Start by graphing the y-intercept (b = 2). • From the y-intercept, apply “rise over run” using your slope.rise = 1, run = -3 • Repeat this again from your new point. • Draw a line through your points. • M = - 1/3 • B = 2 -3 1 -3 1 Start here Y = - 1 X + 2 3 GRAPHING

  7. Intersecting Lines (1,2) • The point where the lines intersect is your solution. • The solution of this graph is (1, 2) GRAPHING

  8. Parallel Lines • These lines never intersect! • Since the lines never cross, there is NO SOLUTION! • Parallel lines have the same slope with different y-intercepts. GRAPHING

  9. Coinciding Lines • These lines are the same! • Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! • Coinciding lines have the same slope and y-intercepts. GRAPHING

  10. SYSTEMS OF EQUATIONS POSSIBLE SOLUTIONS: If you solve using substitution or elimination X and Y can be an ordered pair. X = 4, Y= 7. Answer: (4,7) ONE SOLUTION If you solve, and the variables cancel out, leaving you 8 = 8; This is a true statement therefore, Answer: INFINITELY MANY SOLUTIONS. If you solve, and the variables cancel out, leaving you 8 = 0; This is NOT a true statement therefore, Answer: NO SOLUTION.

  11. Solving a system of equations by substitution Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Step 1: Solve an equation for one variable. Step 2: Substitute Put the equation solved in Step 1 into the other equation. Step 3: Solve the equation. Get the variable by itself. Step 4: Plug back in to find the other variable. Substitute the value of the variable into the equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations. SUBSTITUTION

  12. SOLVE USING SUBSTITUTION 3x – y = 4 x = 4y – 17 SUBSTITUTION

  13. Solving a system of equations by elimination using addition and subtraction. Standard Form: Ax + By = C Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Look for variables that have the same coefficient. Step 3: Add or subtract the equations. Solve for the variable. Step 4: Plug back in to find the other variable. Substitute the value of the variable into the equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations. ELIMINATION

  14. SOLVE USING ELIMINATION x + y = 5 3x – y = 7 ELIMINATION

  15. Solving a system of equations by elimination using multiplication. Standard Form: Ax + By = C Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Look for variables that have the same coefficient. Step 3: Multiply the equations and solve with addition or subtraction. Multiply both equations and solve for the variable. Step 4: Plug back in to find the other variable. Substitute the value of the variable into the equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations. ELIMINATION

  16. SOLVE USING ELIMINATION 2x + 2y = 6 3x – y = 5 ELIMINATION

  17. SOLVE F(X) = G(X) • F(X) = 2X + 4 • G(X) = 6X – 8 • Set the equations equal to each other and solve for x. F(X) = G(X) 2X + 4 = 6X – 8 -2X -2X 4 = 4X – 8 +8 +8 12 = 4X 4 4 3 = X F(X) = G(X)

  18. Slope (M) • A measure of the steepness of a straight line • Tells how fast one variable changes compared with the other. • Rise over run

  19. 3) Find the slope of the line that goes through the points (-5, 3) and (2, 1). SLOPE

  20. Determine the slope of the line. -1 Find points on the graph. Use two of them and apply rise over run. 2 The line is decreasing or going down the hill (slope is negative). SLOPE

  21. Intercept slope Now look at the equation below…… When an equation is in slope-intercept form: y = mx + b 2 What is the slope? ____________ 1 SLOPE What is the intercept? ____________

  22. Find the x- and y-intercepts of x - 2y = 12 x-intercept: Plug in 0 for y. x - 2(0) = 12 x = 12; (12, 0) y-intercept: Plug in 0 for x. 0 - 2y = 12 y = -6; (0, -6) SLOPE

  23. You can also find slope when given a table of values. Pick any two points and find the slope. (1,4) and (2,5) m = y2 – y1 x2 – x1 M = ( 5 – 4 ) ( 2 – 1 ) M = 1 = 1 1 SLOPE

  24. Types of Slope Undefined or No Slope Positive Negative Zero SLOPE

  25. Remember the word “VUXHOY” V=vertical lines U=undefined slope X=number; This is the equation of the line. H=horizontal lines O=zero is the slope Y=number; This is the equation of the line. SLOPE

  26. Nonlinear Functions 12-6 4 x 4 –4 0 –4 Course 2 Tell whether the graph is linear or nonlinear. y A. B. y 4 x 4 –4 0 –4 The graph is a straight line, so the graph is linear. The graph is not a straight line, so it is nonlinear.

  27. Nonlinear Functions Course 2 Tell whether the function in the table has a linear or nonlinear relationship. A. difference = 1 difference = 3 difference = 1 difference = 6 The difference between consecutive input values is constant. The difference between consecutive output values is not constant. The function represented in the table is nonlinear.

  28. Nonlinear Functions 12-6 Course 2 Example 2B: Identifying Nonlinear Relationships in Function Tables Tell whether the function in the table has a linear or nonlinear relationship. A. difference = 1 difference = 3 difference = 1 difference = 3 The difference between consecutive input values is constant. The difference between consecutive output values is constant. The function represented in the table is linear.

  29. Components of a Graph Graphs are used to present numerical information in picture form. • Title • Every graph must have a title • Subtitles • Explains what the horizontal and vertical quantities represent • Equally spaced divisions

  30. Scatter Plot is a graph of two related sets of data on an XY axis. • These are useful when you want to study related pairs, such as height and weight. • Correlation is the relationship between two or more things. • Linear Correlationis a scatter plot that forms a “line” showing that one axis seems to depend on or relate to the other.

  31. Line of Best Fit • Line of best fit- line that seems to describe the direction the points are heading in. • There are methods for determining where this line is, there are two criteria to finding and drawing the line: • The line of best fit must more or less follow the direction of the points. • There should be roughly the same number of points on each side of the line. • Lines of best fit can be used to predict results, especially if you find the line's equation.

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