The following functions are defined for all real values of x. Find the range of the funcion.
1) y=m^2-4m-5
2) y=-9v^2-18v-55.
Thank you for your help. (Please step by step.
they are both parabolas.
#1 opens up, so its range has a minimum
#2 opens down, so its range has a maximum
So, just find the vertex; that defines the closed end of the range.
To find the range of a function, we need to determine the set of all possible output values or the set of all possible y-values.
1) For the function y = m^2 - 4m - 5, we can find the range by considering the behavior of the graph of this quadratic function. However, to simplify the process, we can make use of the fact that this function is a quadratic.
Step 1: Determine the vertex of the quadratic function. The vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic function. In this case, a = 1 (coefficient of m^2) and b = -4 (coefficient of m).
x = -(-4) / (2 * 1)
x = 4 / 2
x = 2
The x-coordinate of the vertex is 2.
Step 2: Substitute the x-coordinate of the vertex into the quadratic function to find the corresponding y-coordinate.
y = (2)^2 - 4(2) - 5
y = 4 - 8 - 5
y = -9
The vertex is (2, -9).
Since the quadratic function opens upward (because the coefficient of m^2 is positive), the vertex represents the minimum point of the function. Therefore, the range of the function is all real values greater than or equal to -9. In set notation, the range can be expressed as {y | y ≥ -9}.
2) For the function y = -9v^2 - 18v - 55, we can follow similar steps as in the previous example.
Step 1: Determine the vertex using the formula x = -b / (2a). In this case, a = -9 and b = -18.
x = -(-18) / (2 * -9)
x = 18 / -18
x = -1
The x-coordinate of the vertex is -1.
Step 2: Substitute the x-coordinate of the vertex into the quadratic function to find the corresponding y-coordinate.
y = -9(-1)^2 - 18(-1) - 55
y = -9 + 18 - 55
y = -46
The vertex is (-1, -46).
Since the quadratic function opens downward (because the coefficient of v^2 is negative), the vertex represents the maximum point of the function. Therefore, the range of the function is all real values less than or equal to -46. In set notation, the range can be expressed as {y | y ≤ -46}.