For the quadratic function : y=-0.25(x+8)(x-6) write the function in the vertex form y=a(x-h)^2+k by completing the square. Show all work.
first, expand the factored form:
y = -1/4 (x^2+2x-48)
now regroup a bit to expose the perfect square:
= -1/4 (x^2+2x+1 - 47)
= -1/4 (x+1)^2 + 47/4
An investment firm hired you to manage one of their new mutual funds. The fund invests in two main companies/stocks, Pittsburg Steel (X1) and IMB(X2). Pittsburg Steel is currently trading at $40.00 per share and IMB at $30.00 per share. Your goal is to maximize the return on investment (ROI) for the initial investment period. Recent market analysis indicates that both companies will be trading higher at the end of this investment period. The ROI for Pittsburg Steel is expected to be $10 per share and the ROI on the IMB stock is expected to be $15 per share. The firm gives you $60,000 as an initial investment to get your fund going but stipulates that at least 500 shares of each stock must be purchased. Additionally your initial research shows that IMB is a higher risk alternative and you decide that no more than one-half of the total number of shares purchased should be invested in IMB.
To write the given quadratic function in vertex form, we will complete the square. The vertex form of a quadratic function is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Step 1: Expand the quadratic expression.
y = -0.25(x + 8)(x - 6)
y = -0.25(x^2 - 6x + 8x - 48)
y = -0.25(x^2 + 2x - 48)
Step 2: Group the x terms.
y = -0.25(x^2 + 2x) - 0.25(48)
Step 3: Compute the value to complete the square.
To complete the square for x^2 + 2x, we need to add (2/2)^2 = 1 to the expression:
y = -0.25(x^2 + 2x + 1) - 0.25(48) + 0.25
Step 4: Simplify the expression inside the parentheses.
y = -0.25(x^2 + 2x + 1) - 12 + 0.25
Step 5: Rewrite the constant terms.
y = -0.25(x + 1)^2 - 11.75
Therefore, the quadratic function y = -0.25(x + 8)(x - 6) can be written in vertex form as y = -0.25(x + 1)^2 - 11.75. The vertex is located at the point (-1, -11.75).