Question

5. Find the vertex form of the following equation

and list the attributes.
y = 3x² + 2x + 5
a) Axis of Symmetry:
b) Vertex:
c) Minimum / Maximum
d) Range:
e) Equation in Vertex Form:

Answer 1

To find the vertex form of the equation y = 3x² + 2x + 5, complete the square, resulting in the vertex form y = 3(x + 1/3)² - 4/3. The attributes of the equation include the axis of symmetry, vertex, minimum/maximum, and range.

Step-by-step explanation:

To find the vertex form of the equation y = 3x² + 2x + 5, we can complete the square. The vertex form is given by y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. In this case, a = 3, so the equation can be rewritten as y = 3(x² + (2/3)x) + 5. To complete the square, we need to add and subtract ((2/3)/2)² = 1/9, so the equation becomes y = 3(x² + (2/3)x + 1/9 - 1/9) + 5. Simplifying this gives us y = 3((x + 1/3)² - 1/9) + 5. Therefore, the vertex form of the equation is y = 3(x + 1/3)² - 4/3. Now, we can determine the attributes of the equation:

1. Axis of Symmetry: The axis of symmetry is given by the equation of the form x = h, where the vertex is (h, k). In this case, the axis of symmetry is x = -1/3.
2. Vertex: The vertex is represented by the coordinates (h, k). In this equation, the vertex is (-1/3, -4/3).
3. Minimum / Maximum: Since the coefficient of the squared term is positive, the parabola opens upwards and the vertex represents the minimum point. Therefore, the equation has a minimum.
4. Range: The range of the equation is determined by the y-coordinate of the vertex. In this case, the range is y ≤ -4/3 (or, in interval notation, (-∞, -4/3]).

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