Question

Consider f(x)=50|x-0.5|+0 Complete the sentences. The function is decreasing over the interval x>0.5 x<0.5 x>-0.5

The vertex is_____(complete the statement with the x-coordinate of the vertex).

Answer 1

The function f(x) = 50|x - 0.5| is decreasing for x > 0.5 and increasing for x < 0.5, with the vertex at x = 0.5.

Step-by-step explanation:

The function given is f(x) = 50|x - 0.5|. To determine the intervals where the function is increasing or decreasing, we can analyze the absolute value component. Absolute value functions have a vertex at the point where the expression inside the absolute value is equal to zero.

In this case, the vertex is at x = 0.5 because that is when |x - 0.5| equals zero.

For values of x less than 0.5, the function is increasing because as x approaches 0.5 from the left, the expression inside the absolute value (x - 0.5) is negative, resulting in a positive slope when multiplied by -50.

For values of x greater than 0.5, the function is decreasing because as x increases beyond 0.5, the expression inside the absolute value becomes positive, and multiplying by 50 maintains a positive slope, yet the overall trend in y values is downward due to the decrease in the value of |x - 0.5|.

Therefore, the function is decreasing over the interval x > 0.5 and it is increasing over the interval x < 0.5.

The vertex of this function, signifying the peak or the lowest point of the V-shape curve, corresponds to the x-coordinate of the vertex, which is x = 0.5.