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Due: 21 October 2011 Math 1090-002 17 October 2011 Sections 3.1 & 3.2 Homework Instructions: Complete the following problems in the space provided. Please show your work; write down enough so that a classmate could follow the steps you’ve taken to arrive at the solution. 1. Consider the equation f (x) = x2 + 5. (a) Identify the vertex of the parabola described by this equation. (2 points) (b) Is the parabola concave up or concave down? Does this mean that the vertex is the maximum point or the minimum point on the curve? (2 points) (c) What is the axis of symmetry? (1 point) (d) Find all real roots of this equation by setting f (x) = 0 and solving for x. (2 points) 1 (e) Plot the parabolas h(x) = x2 and f (x) = x2 + 5 on the axes below. On each curve, plot the vertex as well as the points corresponding to x = 1 and x = −1. (4 points) 2 2. Consider the equation f (x) = −x2 − 14x − 49. (a) Put the above equation into the form f (x) = a(x − h)2 + k by completing the square. (3 points) (b) Identify the vertex of the parabola described by this equation. (2 points) (c) Find all real roots of this equation by setting f (x) = 0 and solving for x. (2 points) 3 (d) Plot the parabolas h(x) = x2 and f (x) = −x2 −14x−49 on the axes below. Plot the vertex and at least two other points on each curve. (You may want to use the points corresponding to x = −6 and x = −8 for f (x).) Also, draw the axis of symmetry for each parabola. 4 (4 points) For the following quadratic equations in standard form f (x) = ax2 + bx + c, complete the square to put them into the form f (x) = a(x − h)2 + k. 3. f (x) = x2 − x + 1. (3 points) i. We don’t have anything to factor out, since a = 1 for this equation. We simply start with f (x) = x2 − x + 1. ii. What constant must we add to x2 − x so that it can be written as (x − 12 )2 ? (Hint: Expand (x − 21 )2 .) iii. Remember that after we add this constant to f (x), we must also subtract it so that we don’t change the expression. Finish the process to arrive at 3 1 f (x) = (x − )2 + . 2 4 5 4. f (x) = −2x2 − 20x + 5. (3 points) (Goal: f (x) = −2(x + 5)2 + 55.) Start by factoring out the −2. 5. f (x) = x2 − 4x + 2. (3 points) (Goal: f (x) = (x − 2)2 − 2.) 6 For the equations in problems 4, 5 and 6 above, find all roots by setting f (x) = 0 and solving for x. Then check your answers by computing the roots again using the quadratic formula. 3. 4. 7 (4 points each) 5. 8 6. Consider the equation f (x) = 2x2 − 10x + 8. (a) Find the roots of f (x) using any method you like. (2 points) (b) Complete the square to put the above equation into the form 9 5 f (x) = 2(x − )2 − . 2 2 (3 points) 9 (c) Plot the parabolas h(x) = x2 and f (x) = 2x2 − 10x + 8 on the axes below. For f (x), plot the vertex and the points (x1 , 0), (x2 , 0) corresponding to the two roots of f (x). (4 points) 10