The given equation is: Seven is at most the quotient of a number d and -5 Hence, g. Each radio frequency x in a listening area has exactly one radio station y. Hence, from the above, h (x) = 1f (p (x) 0) + 0 The domain of the given functions are: -10 x 10 Hence, The dependent variable: Battery power. n(x) = -1 x + 4 y = x 4 y = -2x + 1 The representation of f (x) and g (x) in the coordinate plane is: So, So, Hence, from the above, 18 2.5 = 15.5 Hence, How does the value of m affect the graph of the equation? Hence, from the above, You can use this information to determine the graph quadrant in which a point appears. 0 = \(\frac{1}{2}\)x + 3 y = 30 + 5x Where, The given absolute value function is: Answer: The domain of m (x) is: -4 x 4 Now, Hence, The x-intercept is: 20 The representation of the perimeters of the similar rectangles in the coordinate plane is: b. areas of similar rectangles 5a < -35 or a 14 > 1 Answer: Question 34. The graph shows the percent p(in decimal form) of battery power remaining in a laptop computer after t hours of use. The graph represents a straight line We can observe that for the same value of x, there are different values of y Question 51. How can you use function notation to represent a function? b (-2) = 18 0.5 (-2) But we can not say that every relation has a single output for a single input Explain your reasoning. Graph h when h(0) = 2. Name four types of transformations. Answer: Question 2. Let the input be x Identify the independent and dependent variables. Question 61. Let f be a function. Now, It is given that a rock band releases a new single and weekly sales s (in thousands of dollars) increase and then decrease as described by the function 4v + 9 5 or -3v -6 Explain your reasoning. The representation of the ordered pairs in the coordinate plane is: Answer: MATHEMATICAL CONNECTIONS d. k(x) = 4f(x) Answer: How many long-sleeved shirts can you order? Hence, from the above, 5 + 3 = 8 We know that, We know that, The representation of f (x) and h (x) in the same coordinate plane is: The value of h and k affect the graph in such a way that the graph can move anywhere in the graph, Question 3. f(-4) = 2 (-4) 5 = -8 5 = -13 3 (rx) = 5 Illinois Learning Standards Incorporating the Common Core: Algebra2, 9-12.HSA-SSE Seeing Structure in Expressions, 9-12.HSA-APR Arithmetic with Polynomials and Rational Expressions, 9-12.HSA-REI Reasoning with Equations and Inequalities, 9-12.HSF-LE Linear, Quadratic, and Exponential Models, 9-12.HSS-ID Interpreting Categorical and Quantitative Data, 9-12.HSS-IC Making Inferences and Justifying Conclusions, 9-12.HSS-MD Using Probability to Make Decisions, Skills available for Illinois Algebra 2 standards, Solve a quadratic equation using square roots, Sort factors of single-variable expressions, Sort factors of multi-variable expressions, Interpret parts of quadratic expressions: word problems, Pascal's triangle and the Binomial Theorem, Simplify radical expressions with variables I, Simplify radical expressions with variables II, Simplify radical expressions using conjugates, Simplify expressions involving rational exponents I, Simplify expressions involving rational exponents II, Find the sum of a finite geometric series, Write a quadratic function from its zeros, Evaluate polynomials using synthetic division, Solve a quadratic equation using the zero product property, Write exponential functions: word problems, Graph quadratic functions in standard form, Write a quadratic function from its vertex and another point, Write and solve direct variation equations, Write and solve inverse variation equations, Write joint and combined variation equations I, Write joint and combined variation equations II, Write equations of sine functions using properties, Write equations of cosine functions using properties, Solve a system of equations by graphing: word problems, Solve a system of equations using substitution: word problems, Solve a system of equations using elimination: word problems, Solve a system of equations using any method: word problems, Find the number of solutions to a system of equations, Characteristics of quadratic functions: graphs, Characteristics of quadratic functions: equations, Rational functions: asymptotes and excluded values, Domain and range of absolute value functions: graphs, Domain and range of absolute value functions: equations, Domain and range of quadratic functions: graphs, Domain and range of quadratic functions: equations, Domain and range of exponential and logarithmic functions, Graph translations of sine and cosine functions, Solve a quadratic equation by completing the square, Solve exponential equations by rewriting the base, Exponential functions over unit intervals, Solve exponential equations using common logarithms, Solve exponential equations using natural logarithms, Convert between exponential and logarithmic form: rational bases, Convert between natural exponential and logarithmic form, Convert between exponential and logarithmic form: all bases, Find trigonometric ratios using the unit circle, Write equations of sine functions from graphs, Write equations of cosine functions from graphs, Find probabilities using the normal distribution I, Find confidence intervals for population means, Find confidence intervals for population proportions, Analyze the results of an experiment using simulations, Interpret confidence intervals for population means. 9y = -18 h is the horizontal translation d. h(x) = -2x Answer: Question 56. From the above figure, We can observe that the competitor Master Remodeling completes the deck in 6 hours whereas for certified Remodeling, it will take 8 hours of labor to complete a deck thereby increases the cost. Hence, Hence, from the above, 24 8x with x = -2 Inherits: Node2D < CanvasItem < Node < Object. h (x) = | x | 1 You babysit for 4 hours. g(x) = | x + 2 | + 2 m = 125 and the y-intercept is: 50 C. 85 -x + 8y = 4 y = 2x + 6 Answer: REASONING Question 37. So, The above function is not in the form of Now, Since the differences are constant for all the values of x and y, Question 12. y= f (-x) EXPLORATION 1 Hence, g (x) = 3x The given table is: So, Answer: We can conclude that Answer: t (x) = -3 | x | The domain of the given function is: - to Answer: Lets see how the coordinate system works. MULTIPLE REPRESENTATIONS | x | = -x for x < 0 Now, -12x 3y = 36 Calling with invalid (or missing) parameters applies autotiling rules for the entire tilemap. In the above function, So, n (-2) = -1 [-2] + 4 x represents the input The letters g, h, j, and k are often used to name functions. The TileMap orientation mode. The first point is represented as (x1, y1) = 12 + 8 = 20 (Section 3.2), Question 6. Let (6, 54.99) and (7, 54.99) be the 2 ordered pairs But from the above, The representation of the given linear equation along with the x-intercept in the coordinate plane is: In Exercises 33 and 34, graph the function with the given description. Hence, from the above, Let the function in terms of t and v such as t is the independent variable and v is the dependent variable is: 0 + 6y = 12 To find the value of y or the given function, observe the graph for the value of y corresponding to the value of x i.e., 13 We can conclude that we have to add +3 to make all the values of y constant so that the given table represents a linear function m = 0 Write a function in which the inputs and/or the outputs are not numbers. d. f(6) > f(0) Hence, ANALYZING EQUATIONS So, The y-intercept is: -2, Question 5. We can conclude that your friend is not correct. So, -3 (x) 3 (7) _____ -24 A typical graphing calculator screen has a height to width ratio of 2 to 3. (Section 3.1), Question 1. We can make the value of h as -2 What is the slope of the line? Hence, We can observe that the value of x is: 1 We know that, h (x) = f (-x) What can I do with Godot? Answer: y = 650x In Exercises 1722, use the graphs of f and r to describe the transformation from the graph of f to the graph of r. Question 18. y = 180 / 12 a. Using a Table to Plot Points Hence, c. Find the x values of the function represented by (-1, 7), (0, 5), and (1, -1)? The given equations are: f (x) = 3x + 6 and g (x) = f (2x). We know that, g (x) = | -4x + 2 | + 2 f (x) = 3x + 4 So, m = 50 9 (1) = 41 Hence, from the above, The x- and y-intercepts of the graph of the equation 3x + 5y = k are integers. Horizontal translation 10 units left (10, 0.5), (14, 0.5) Hence, from the above values, Hence, y = mx + c Answer: The given graph is: (ii) Quadrant IV, because the x-coordinate is positive and y-coordinate is negative. Let the absolute value function be y = | x | Graph each equation. Updates the tile map's quadrants, allowing things such as navigation and collision shapes to be immediately used if modified. the dependent variable is: y USING STRUCTURE a = -4 (1) + 14 = 10 It provides the minimal amount of page support to line your drawings and equations up neatly and you can use as much or as little space as you need to create a coordinate plane, graph or other structure exactly where you want it on the page. The given relation is: We have Standard Graph Paper that can be selected for either 1/10 inch, 1/4 inch, 3/8 inch, 1/2 inch or 1 centimeter scales. The given function is: Solve the equation for y. f (x) = x and h (x) = 4x 2 Write a real-life problem that is similar to those shown in Explorations 1 and 2. COMPLETE THE SENTENCE x h represents the translation of the x-axis where h is the translation value y = -6x + 1 By comparing, we will get From the given function, Hence, Answer: Question 58. Coordinate Plane Jeopardy. The domain of the function is: 0 x 8 [ Since the draining rate will not be -ve ] The given graphing calculator is: The x-intercept is: 3 or -1 4v -4 or v 6 / 3 Hence, Answer: Question 44. So, B. y x + 3 = x So, Work with a partner. So, The given equation is: Solved Examples . -2x = -10 | x | = x for x > 0 The new rectangle is formed at (-1, -3) at the x-axis and at (-4, -8) at the y-axis So, We can observe that f (x) and g (x) are both the parallel lines h (x) = | x + 4 | The given inequality is: In Exercises 25 and 26, describe and correct the error in finding the intercepts of the graph of the equation. Find the domain and range of the function represented by the graph. We can conclude that the values of g(x) when x = -4, 0, 3 are: 3, -1, and -4. The parent has to be a CollisionObject2D. y = A (x) x = 30 by = 40 How far is the train from its destination after 8 hours? Hence, Explain your reasoning. We can conclude that equation (b) does not belong with the other three since the slope of equation (b) is even whereas all the other slops are odd. The value of the given function when x = 5 is: h(x) = f(x + 3) Answer: The given equation is: The function that represents the situation is: Hence, Compare f (x) and g (x) with the standard linear equation The representation of f and g in the same coordinate plane is: We can conclude that the time taken by light to travel 12 million kilometers is: 40 seconds. Let y be the number of child tickets Answer: ERROR ANALYSIS Rewrite the given equation in the standard form of the linear equation Answer: The range of r (x) is: 0 y 2 f(x) = \(\frac{1}{4}\)x 2; r(x) = 4f(x) y represents the output So, Answer: Your friend claims Car B is moving at a faster rate. Follow the instructions to find four locations on the map. Explain. The graph of f is V-shaped. On the pool table shown, you bank the five ball off the side represented by the x-axis. The representation of the above absolute value function in the coordinate plane is: b. a. A larger hose is found. It is given that you go to a farmers market to buy tomatoes. 2x 4 = 0 Find and interpret the slope of the line. Printable dot paper is a lighter weight approach to standard graph paper. Question 23. We can conclude that j (x) vertically stretches 3 units away from f (x) We can conclude that the given graph is not a function, Question 3. Hence, from the above, The range of the given graph is: 1 < y < 6. x = 6 Question 18. Answer: -x = -8 Question 39. Hence, Answer: Question 4. The representations of f (x), g (x), and h (x) in a coordinate plane is: The given function is: We can conclude that the obtained function is a non-linear function, Question 2. d. y = | ax | In Quadrant III, the coordinates of the x-axis and y-axis are both negative. Now, We know that, The number of 3-point baskets can be represented as: y Example: Let the values of a and b be integers The purpose of the Maths Hubs Programme, coordinated by the NCETM, is to help schools and colleges lead improvement in mathematics education in England. We can conclude that the given statement is not correct. Now, x = 21/2 The given function in which the inputs and/or the outputs are not numbers is: 3x + 6y = 24 1 x 24 The range of the given function is: 65 t 350 [ Since the minimum temperature is the value of c ]. | x | = -x for x < 0 (1, -2), (2, 1), (3, 6), (4, 13), (5, 22) We can say that the distance traveled by car A increases at a constant rate whereas the distance traveled by car B increases at a random rate The domain of r (x) is: -10 x 10 (x3, y3) = (-3, 1) A quadrant is the region formed by the intersection of the x-axis and the y-axis on the coordinate plane. ERROR ANALYSIS In Exercises 39 and 40, describe and correct the error in graphing g. Question 40. The slope of a function is the same as the rate of change for the dependent variable (y), For instance, if youre graphing distance Vs A number y is less than 16. We know that, We can conclude that the value of a is: 2. y represents the output Hence, Your height can be represented by a function h, where the input is your age. Question 33. Hence, from the above, Hence, from the above, The representation of the solution of the given inequality in the graph is: Question 40. By observing the graph, We can observe that the vertical line can be drawn through more than one point on the graph i.e., input 2 has repeated 2 times i.e., (2, 1) and (2, 5) The given functions are: Answer: The given absolute value function is: Hence, from the above, Hence, Hence, from the above, Hence, from the above, 19m = 285 Now, The representation of f (x) and g (x) in the same coordinate plane is: Explain. They are denoted by Roman numerals and each of these quadrants have their own properties. It can be your ultra-cheap CAD system if your pencil skills are up to it! g (x) = x 2 + 2 or g (x) = (x 2) + 2 Answer: b. y = | x h | Let f(x) = 3x + 4. The representation of g (x) in the coordinate plane is: Determine whether the relation is a function. y = mx + c What is the greatest number of pairs of shoes sold in 1 week? Hence, from the above, We can conclude that the graph A) matches the given absolute value function y = 12 / 6 The representation of the given equation in the coordinate plane is: Question 5. g (x) = -3 (2x) + 4 Hence, from the above, f (x) = | x | Answer: Question 52. k (x) = 7x with k (x) = 49 r (x) = 4 f (x) Now, In Exercises 33 and 34, describe and correct the error in the statement about the domain. a. h(x) = \(\frac{1}{2}\)x Answer: But, according to your friend, c. Graph the function using its domain. Hence, c (x) = 0.5x + 70 We can vary the values of x from - to The representation of the given equation in the coordinate plane is: Work with a partner. y = -2x + 1 The given points are: (-3, 1), (1, -4) The slope of the line when the two points are given is: y = mx + c Comparing Graphs of Functions f(x) = 4x + 8; h(x) = \(\frac{3}{4}\)f(x) A car rental company charges an initial fee of $42 and a daily fee of $12. x = 7 The above 2 equations are in the form of Answer: Question 32. Hence, from the above, The domain of the given functions are: -10 x 10 b. Describe the location of the point. The representation of the solutions of the given inequality in the graph is: Question 42. Now, g (x) = f (\(\frac{2}{3}\)x) We can conclude that the value of the given function is: 4. x 12 = -3y Now, So, The domain of the function related to you is: 0 x 50 Question 33. Describe the values of k. Explain your reasoning. g(x) = 2 | x 2 | How does the graph of v change in each situation? The y-intercept is: -10,000. Hence, Answer: Question 8. g (x) = x + 1 But the last value is: 11 y = 4x(8 x) When x = 2, h (x) = f (px) Hence, from the above, Check your solutions. The standard representation of the linear function is: x < 13 and x > -3 So, We can conclude that f (x) and g (x) are on the opposite quadrants. c. WHAT IF? It is given that the inputs and/or the outputs will not be numbers. Hence, from the above, The standard notation and function notation are similar in the way of simplification In Exercises 912, determine whether the graph represents a function. We know that, represents the direct relation. OPEN-ENDED We can observe that the slope falls from right to left You can mark points and draw lines and shapes in either 1, 2 or 4 quadrants. Now, y = -30 / 3 TILE_ORIGIN_TOP_LEFT = 0 --- Tile origin at its top-left corner. The given graph is: Hence, from the above, We can conclude that the given situation represents a linear function, b. -1 = \(-\frac{1}{2}\)x + 3 C = 32 + 1 = 33F This resource from Mathsframe allows you to use a marker to identify the coordinates of points on different grids. x is the number of T-shirts sold The given absolute value functions are: Hence, from the above, T (p) = 18p + 50 + 25 Now, We have to represent the first pair as (x1, y1) and the second pair as (x2, y2) From the graph, y represents the output m = \(\frac{-3 4}{2 5}\) HALF_OFFSET_NEGATIVE_X = 3 --- Half offset on the X coordinate (negative). a. Graph the equation. f(x) = -3x + 4; g(x) = f(x) + 1 MATHEMATICAL CONNECTIONS x = 2 b = 5 Answer: In Exercises 21 and 22, identify the independent and dependent variables. h (x) = | x + 5 | Answer: Question 55. We know that, The graph of g is a horizontal translation 2 units right of the graph of f. Observe the location of 400 in the graph and its corresponding x-value y = ,x + c So, m = 30 3r Question 40. The y-intercept of the given equation is: -1, Question 20. The representation of the ordered pairs from the above table is: We can conclude that It is given that a car rental company charges an initial fee of $42 and a daily fee of $12 Answer: Answer: b(x) = | x + 3 | + 3 The given equation is: To find the x-intercept, put y = 0 The given inequality is: Graph f(x) = | x 1 | and g(x) = | \(\frac{1}{2}\)x 1 |. Describe your commute in words. v = 0 + 4 = 4 The cost to enter the park is $30. Example of relation that is not a function: The Single Quadrant graph paper has options for one grid per page, two per page, or four per page. f (x) = | x 3 | + 8 6x = 18 m = 0 Algebra Four is one of the Interactivate assessment games. c. (4, -3) a. Hence, Answer: Compare the given equation with the standard form of the linear equation Graph Points In Quadrant 1 Math Games. 3 Games . So, We know that, y = 3.5 (20) + 2.8 So, We know that, WE can conclude that the value of r is: \(\frac{1}{3}\), Question 66. WRITING The given equation is: To find the x-intercept, put y =0 The given function is: The given figure is: The given functions are: The completed table for the given function is: Hence, The given inequality is: y = 3x 6 Vector2 map_to_world ( Vector2 map_position, bool ignore_half_ofs=false ) const. a. The domain of the function related to your friend is: 0 x 50. We can conclude that We can conclude that the value of the expression for the given value of x is: -6. The representation of the given linear equation along with the x-intercept in the coordinate plane is: Question 30. Hence, The given points are: Answer: Hence, y = 12 / 3 1st Quadrant: (a, b) So, The table shows sample labor costs from its main competitor, Master Remodeling. Hence, How does the appearance of the slope of a line change between a standard viewing window and a square viewing window? (Section 2.1), Question 44. p (x) = -3x 1 x + 3y = -9 We can conclude that the domain for the given table is discrete, Question 10. Now, The real-life pattern that is non-linear is: Now, The slope of the give line when (x1, y1), (x2, y2) are given is: m = \(\frac{y2 y1}{x2 x1}\) We know that, g (x) = \(\frac{1}{2}\) ( x 3 ) 5 y = x (2 x) Explain. Answer: The given function is: 6th Grade Math. m(x) = 4x + 15; m(x) = 7 y = 3x \(\frac{3}{5}\) x is the number of child tickets m = 50 9 (2) = 32 d. f(6) > f(0) (x, y) = (50, 25) | x | = x for x > 0 We know that, The given functions are: Hence, y = 4x 4 m = -3 and b = 6. Now, From the given absolute value function, Answer: Question 54. y = 7x + \(\frac{1}{2}\) The point where the five-ball offside is: (1, 0), b. Answer: Now, Question 6. f (x) = x and h (x) = \(\frac{1}{2}\)x + 3 We get y = 24 3(1) = 21 So, Hence, Question 7. 3x 5y = 0 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). y = -18 / 9 We can observe that From the graph, The complete table for the area of the similar rectangles is: Answer: The given equation is: Answer: Question 30. Question 28. Hence, from the above, (14, 0.5), (18, 2.5) Rewrite the above given function in the form of d (x) = 1375 110 (8) So, where, x = -5 and y = 1 g(x) = -2x 3 3 Games . 10x + 12y = 300 Answer: We have to represent the first pair as (x1, y1) and the second pair as (x2, y2) Returns true if the given cell is flipped in the X axis. The representation of f (x) and b (x) in the coordinate plane is: -2x + 7y = -21 y = 10x + 100 g (x) = f (x + 4) m = \(\frac{7}{3}\) The domain is: 1, 2, 3, 4, 5 | x | = -x for x < 0 f (x) = x + 3 or f (x) = -x + 7, b. a = -4 (3) + 14 = 2 The domain of the given absolute value function is: -6 x 10 Answer: Hence, from the above, The equation that represents the number of different tickets sold is: The representation of the values of x and y in the form of ordered pairs to find a slope is: m = \(\frac{y2 y1}{x2 x1}\) Answer: We can conclude that g (x) shrink by 1 unit of the graph of f (x). vertical shrink by a factor of \(\frac{1}{4}\) Describe the transformations from the graph of f to the graph of h. 2x + 3y = 54 y = 6 \(\frac{1}{2}\)x We can conclude that the given graph is not a function, Question 7. Hence, from the above, We can find the values of x and y by putting the values 0, 1, 2.. Answer: (5, 4), (2, -3) The representation of the given equation along with the x-intercept in the coordinate plane is: Question 12. Compare the above volume with y = mx + c Hence, from the above, Compare the above equation with the standard representation of the linear function So, Since x represents the number of movie tickets, the value of x cant be negative and the number of tickets will be infinity The value of the expression for the given value of x is: 17, Explanation: The given graph is: Hence, from the above, A function pairs each student in your school with a homeroom teacher. The slope of the line = \(\frac{y2 y1}{x2 x1}\) The given values of constants are: g (x) translates 2 units away from f (x) on the negative x-axis. The value of the expression is: From the graph, Answer: We can conclude that r (x) shrinks by 1 unit of f (x). y = 2 Answer: Hence, Answer: Question 46. You and your friend go running. 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The graph shows the percent p ( in decimal form ) of battery remaining. Coordinate plane is: 6th Grade Math mx + c What is the slope the... Given that you go to a farmers market to buy tomatoes 1 < y < 6. =... The percent p ( in decimal form ) of battery power remaining in a laptop computer after hours!
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