Below, there are four functions. Select the functions with a constant rate of change.
(2 points)
In order to determine which functions have a constant rate of change, we need to determine if the functions have the same rate of change for any given interval.
A constant rate of change means that for any interval on the graph, the ratio of the change in the dependent variable (usually denoted as y) to the change in the independent variable (usually denoted as x) remains constant.
Let's examine each function:
1) y = 3x + 2
This is a linear function in the form of y = mx + b, where m is the coefficient of x. The coefficient of x, which is 3 in this case, determines the rate of change. Since it is constant, this function has a constant rate of change.
2) y = x^2 + 4
This is a quadratic function. The rate of change for a quadratic function is not constant. It depends on the value of x.
3) y = 5x^3 + 2x
This is a cubic function. Similar to quadratic functions, cubic functions do not have a constant rate of change. The rate of change depends on the value of x.
4) y = sqrt(x)
This is a square root function. Like quadratic and cubic functions, square root functions do not have a constant rate of change.
Therefore, the function with a constant rate of change is y = 3x + 2.
To determine which functions have a constant rate of change, we need to look for functions that have a consistent change in output for each unit change in input. In other words, the slope or gradient of the function should remain constant.
Let's examine each function:
1. f(x) = 2x + 3: This function is a linear function with a constant rate of change. The coefficient of x, which is 2, represents the rate of change or slope and remains constant. Thus, this function has a constant rate of change.
2. g(x) = x^2 + 3: This function is a quadratic function with a non-constant rate of change. The rate of change varies based on the value of x. Therefore, this function does not have a constant rate of change.
3. h(x) = 5: This function is a constant function, which means the output is always the same regardless of the input. It has a rate of change of zero since the output does not change with any change in input. Therefore, this function has a constant rate of change.
4. k(x) = sin(x): This function is a trigonometric function that oscillates between -1 and 1 in a wave-like pattern. The rate of change or slope of this function varies at different points since it is not a straight line. Therefore, this function does not have a constant rate of change.
In summary, the functions with a constant rate of change are f(x) = 2x + 3 and h(x) = 5.
To determine which functions have a constant rate of change, we need to examine each function and see if the rate of change remains the same for every input.
1. f(x) = 3x + 2:
- This is a linear function with a slope of 3. It has a constant rate of change since the slope remains the same for every input.
2. g(x) = x^2 - 1:
- This is a quadratic function. The rate of change is not constant as it depends on the specific input value.
3. h(x) = 5cos(x):
- This is a trigonometric function. The rate of change is not constant as it varies based on the cosine function.
4. k(x) = 2^x:
- This is an exponential function. The rate of change is not constant as it depends on the exponent of 2.
Therefore, the only function with a constant rate of change is f(x) = 3x + 2.