Identify the maximum or minimum value and the domain and range of the graph of the function y = 2(x – 3)2 – 4. (1 point) Responses minimum value: –4 domain: all real numbers range: all real numbers –4 minimum value: –4 domain: all real numbers range: all real numbers Image with alt text: greater than equal to –4 maximum value: 4 domain: all real numbers range: all real numbers 4 maximum value: 4 domain: all real numbers range: all real numbers Image with alt text: A symbol less than or equal to 4 maximum value: –4 domain: all real numbers –4 range: all real numbers maximum value: –4 domain: all real numbers Image with alt text: A symbol less than or equal to –4 range: all real numbers minimum value: 4 domain: all real numbers 4 range: all real numbers
The correct response is:
Minimum value: -4
Domain: all real numbers
Range: greater than or equal to -4
The correct response is:
- The minimum value is -4.
- The domain is all real numbers.
- The range is y ≤ -4.
Image with alt text: The graph is a parabola that opens upward, with the vertex at (3, -4) and the axis of symmetry is x = 3. The graph extends infinitely in both directions and is bounded below by the y-coordinate of the vertex.
To identify the maximum or minimum value and the domain and range of the graph of the function y = 2(x – 3)² – 4, you can follow these steps:
1. Start by rewriting the function in vertex form: y = a(x - h)² + k.
In this case, a = 2, h = 3, and k = -4. So the vertex form of the function is y = 2(x - 3)² - 4.
2. The vertex form of a quadratic function (in this case, a parabola) is useful for identifying the maximum or minimum value. The vertex of the parabola represents the maximum or minimum point.
In this case, since a = 2 is positive, the parabola opens upward. Therefore, the vertex represents the minimum point.
3. To find the vertex, use the formula h = -b/(2a) to find the x-coordinate and substitute it into the function to find the y-coordinate.
In this case, the x-coordinate of the vertex is h = -(-3) / (2*2) = 3/4.
Substituting x = 3/4 into the function, we get:
y = 2(3/4 - 3)² - 4
y = 2(-9/4) - 4
y = -9/2 - 4
y = -9/2 - 8/2
y = -17/2
4. Therefore, the vertex of the parabola is (3/4, -17/2), and the minimum value is -17/2.
5. The domain of the function is all real numbers, as there are no restrictions on the x-values.
6. For the range, notice that the parabola opens upward, which means the range extends upward indefinitely. However, the vertex represents the minimum value, so the range is from the minimum value (-17/2) to positive infinity.
So, the correct response is: minimum value: -17/2, domain: all real numbers, range: from -17/2 to positive infinity.