Quadratic function and equation #3 - Questions and Answers

Question 1 One of the two tables below shows data that can best be modeled by a linear function, and the other shows data that can best be modeled by a quadratic function. Identify which table shows the linear data and which table shows the quadratic data, and find a formula for each model. Table A | x | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | f(x) | 8 | 21 | 42 | 71 | 108 | Table B | x | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | g(x) | 8 | 21 | 34 | 47 | 60 | linear function = quadratic function = Answer Step 1 Given Table A | x | f(x) = y | x² | x³ | x⁴ | xy | x²y | |---|---|---|---|---|---|---| | 0 | 8 | 0 | 0 | 0 | 0 | 0 | | 1 | 21 | 1 | 1 | 1 | 21 | 21 | | 2 | 42 | 4 | 8 | 16 | 84 | 168 | | 3 | 71 | 9 | 27 | 81 | 213 | 639 | | 4 | 108 | 16 | 64 | 256 | 432 | 1728 | | 10 | 250 | 30 | 100 | 354 | 750 | 2556 | y = a + bx + cx² quadratic equation normal equation are ∑y = an + b∑x + c∑x² ∑xy = a∑x + b∑x² + c∑x³ ∑x²y = a∑x² + b∑x³ + c∑x⁴ Substituting there value in above equation 5a + 10b + 30c = 250 10a + 30b + 100c = 750 30a + 100b + 354c = 2556 (1) * 2 - (2) -10b - 40c = -250 (1) * 6 - (3) -40b - 174c = -1056 (4) x 4 - (5) -14c = -56 c = 4 From (3) -10b - 40c = -250 -10b - 160 = -250 Step 2 b = -90 / -10 = 9 From (1) 5a + 90 + 120 = 250 5a = 40 a = 8 Put a, b, and c in quadratic equation we get f(x) = y = 8 + 9x + 4x² b) Given Table B | x | g(x) = y | x² | x * y | |---|---|---|---| | 0 | 8 | 0 | 0 | | 1 | 21 | 1 | 21 | | 2 | 34 | 4 | 68 | | 3 | 47 | 9 | 141 | | 4 | 60 | 16 | 240 | | 10 | 250 | 30 | 470 | Straight line equation y = a + bx normal equation are ∑y = an + b∑x ∑xy = a∑x + b∑x² linear function = Table B quadratic function = Table A Question 2 Write a quadratic equation having the given solutions. -6, 9 Write the quadratic equation in standard form, ax² + bx + c = 0 where a is positive, and a, b, and c are integers with no common factor. The quadratic equation is nothing. Answer Step 1 Since -6, 9 be the solution of the quadratic equation so (x + 6)(x - 9) = 0 ⇒ x² - 6x + 9x - 54 = 0 ⇒ x² + 3x - 54 = 0 Hence the quadratic equation F(x) = x² + 3x - 54 = 0 Question 3 Compute the value of the discriminant and give the number of real solutions of the quadratic equation -3x² + 6x-7=0 Answer Given -3x² + 6x - 7 = 0 We must calculate the discriminant's value and identify several viable solutions. The discriminant's value is determined by D = b² - 4ac Where D is discriminant here a = -3. b = 6. c = -7 So, D = b² - 4ac D = (6)² - 4(-3) (-7) D = 36 - 4(21) D = 36 - 84 D = -48 Discrimination is harmful and has no answers. The discriminant is -48, and there are no meaningful solutions. Question 4 Given the graph of g below: a. Determine the family of functions to which g belongs (linear, quadratic, exponential, etc.) and write the equation for the parent function, f. b. List the transformations that need to be performed on the graph of f, in the correet order to obtain the graph of g. c. Write the equation of g. a) Family: b) Transformations c) Equation of g: [Image of the graph was provided and described in the text] Answer Step 1: Given, The graph of the function g(x). We have to answer the following... Step 2: Explanation Solution (a): The given function is not a linear function. This is a quadratic function. Also, the parent function for the given graph is f(x) = |x|. Solution (b): Since we have the function f(x) = |x|, then Take a reflection of the function over the x-axis, we get f(x) = -|x|. Take a horizontal translation left 3 units, we get f(x) = -|x - 3|. Take a vertical translation up 2 units, we get g(x) = -|x - 3| + 2. Solution (c): The equation of the graph g(x) is g(x) = -|x - 3| + 2.