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# What is the property of the function y = (1/2)x2 + 3x + 1/2?

Question:

The function of a parabola is y = (1/2)x2 + 3x + 1/2. Answer the following questions: 1. What is vertex coordinate and symmetry axis of the function? 2. What are the coordinates of the graph of y = (1/2)x2 + 3x + 1/2 intersecting with x-axis? 3. Skitch the graph.

Solution:

Answer of question 1: To find the vertex coordinate and symmetry axis of the function y = (1/2) x2 + 3x + 1/2, we need applying the completing square formula to change the given general form of the parabola into the vertex form of the parabola. y = (1/2) x2 + 3x + 1/2 = (1/2) (x2 + 6x) + 1/2 = (1/2) (x2 + 6x + 32 - 32) + 1/2 = (1/2) (x2 + 6x + 32) - 9/2 + 1/2 = (1/2) (x + 3) 2 - 4. Therefore, the vertex coordinate of the parabola is (-3, -4). The x-coordinate of the vertex is -3 and y-coordinate of the vertex is -4.

Answer of question 2: When the graph of the function y = (1/2) (x + 3) 2 - 4 intersects with x-axis, y = 0. Substitute y = 0 into the equation, we get (1/2) (x + 3) 2 - 4 = 0. Both sides of the equation add 4, then (1/2) (x + 3) 2 = 4. Both sides of the equation times 2, then (x + 3) 2 = 8. Square both sides of the equation, then x + 3 = + - square root of 8. Thus, we get two solutions: x1 = -3 + 2 (square root of 2) = -0.17 and x2 = -3 - 2 (square root of 2) = -5.83. Therefore, x1 = -0.17 and x2 = -5.83. So, the graph intersects with the x-axis in two points P1 and P2. P1 has the coordinate (-0.17, 0) and P2 has the coordinate (-5.83, 0).

Answer of question 3: Because the coefficient of the x square term is a positive number, so the graph of the function y = (1/2) x2 + 3x + 1/2 is upward. Look the video to see the graph.