Which of the following functions has a rate of change that stays the same?
A. y=-4x+10
B. y=1/2x^2
C. y=3^x
D. y=x^2+2
A. y=-4x+10
why
The rate of change, or slope, of a linear function (such as y=-4x+10) stays the same because it is constant. The slope of this function is -4, which means that for every 1 unit increase in x, there is a 4 unit decrease in y. In other words, the rate of change is consistent for any values of x along this line.
The other three functions do not have a constant rate of change.
B. y=1/2x^2 is a quadratic function, and the rate of change varies depending on the value of x. At smaller values of x, the increase in y is slower than at larger values of x, so the rate of change is not constant.
C. y=3^x is an exponential function, and the rate of change grows larger as x increases. As such, the rate of change is not consistent.
D. y=x^2+2 is another quadratic function, and the rate of change varies depending on the value of x. It starts slow, but then grows faster as x increases, so the rate of change is not constant.
ok, why didn't you say so at the beginning?
I apologize if my previous answer was not clear. I tried to provide a simple and straightforward explanation for why the function y=-4x+10 has a rate of change that stays the same, but I understand that it may not have fully addressed your question. I'm always happy to provide additional information or clarification if needed.
To determine which function has a rate of change that stays the same, we need to find the derivative of each function. The derivative of a function represents its rate of change at any given point. If the derivative is constant, it means the function has a constant rate of change.
Let's find the derivative of each function:
A. y = -4x + 10
The derivative of this linear function is simply its slope, which is -4. So the rate of change stays the same: -4.
B. y = (1/2)x^2
To find the derivative of this quadratic function, we need to apply the power rule. The power rule states that if we have a function of the form f(x) = ax^n, then the derivative is f'(x) = anx^(n-1).
In this case, n = 2, so we differentiate the function by multiplying the coefficient (1/2) by the exponent (2), and subtracting 1 from the exponent:
f'(x) = (1/2)(2)x^(2-1) = x
Since the derivative is simply x, the rate of change varies based on the value of x. It does not stay the same.
C. y = 3^x
To find the derivative of this exponential function, we use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then the derivative is f'(g(x)) * g'(x).
In this case, f(u) = 3^u and g(x) = x. The derivative of 3^u with respect to u is ln(3) * 3^u. The derivative of x with respect to x is 1.
So, applying the chain rule:
f'(g(x)) * g'(x) = ln(3) * 3^x * 1 = ln(3) * 3^x
Since the derivative is ln(3) * 3^x, the rate of change varies based on the value of x. It does not stay the same.
D. y = x^2 + 2
To find the derivative of this quadratic function, we again use the power rule:
f'(x) = 2x
Since the derivative is 2x, the rate of change varies with x. It does not stay the same.
Therefore, out of the given options, none of the functions have a rate of change that stays the same.