Question

Find the intercepts, vertex, max or min, and range. g(x)=-(x+2)²+3

Answer 1

To find the intercepts, vertex, maximum or minimum, and range of the function g(x) = -(x+2)² + 3, use the vertex form of a quadratic equation, y = a(x-h)² + k. The x-intercept is -2, the y-intercept is -1. The vertex is (-2, 3), and the function has a maximum at the vertex. The range is [-∞, 3].

Step-by-step explanation:

To find the intercepts, vertex, maximum or minimum, and range of the function g(x) = -(x+2)² + 3, we can use the vertex form of a quadratic equation, y = a(x-h)² + k, where (h,k) represents the vertex. In this case, the equation has a negative coefficient, indicating the vertex is a maximum point.

1. x-intercept: To find the x-intercept, set the function equal to zero and solve for x:
   0 = -(x+2)² + 3. Solving this equation, we get x = -2.
2. y-intercept: To find the y-intercept, substitute x = 0 into the function:
   g(0) = -(0+2)² + 3 = -4 + 3 = -1. Therefore, the y-intercept is -1.
3. Vertex: The vertex of the function is the point (h,k).
4. h = -2 (from the equation).
5. k = 3 (from the equation).
6. Maximum: Since the coefficient of the squared term is negative, the function has a maximum point at the vertex.
7. Range: The range of the function is all real numbers less than or equal to the y-coordinate of the vertex. In this case, the range is [-∞, 3].

Therefore, the intercepts, vertex, maximum or minimum, and range of the function g(x) = -(x+2)² + 3 are:

• X-intercept: -2
• Y-intercept: -1
• Vertex: (-2, 3)
• Maximum: 3
• Range: [-∞, 3]