Question

Построить график функции y=(x-4)^2 -1 указать область определения, нули функции, промежутки знакопостоянства, промежутки возрастания и убывания, область изменения функции

Answer 1

У=(х-2)²-1
Графиком является парабола у=х² с вершиной в точке (2;-1),которая является точкой минимума.
D(y)∈(-∞;∞)
Нули функции  (0;3) (1;0) (3;0)
y>0  (-∞;1) U (3;∞)
y<0    (1;3)
Возр (2;∞) и убыв (-∞;2)
E(y)∈[2;∞)

Answer 2

Graph of the Function

The function is given by y = (x-4)^2 - 1. Let's start by plotting the graph of this function.

The graph of the function y = (x-4)^2 - 1 is a parabola that opens upwards, with the vertex at (4, -1).

Domain of the Function

The domain of the function is all real numbers, as there are no restrictions on the values that x can take.

Zeros of the Function

To find the zeros of the function, we set y = 0 and solve for x: 0 = (x-4)^2 - 1 (x-4)^2 = 1 x-4 = ±1 x = 4 ± 1 x = 3, 5

So, the function has zeros at x = 3 and x = 5.

Intervals of Constant Sign

To determine the intervals of constant sign, we can use the zeros of the function to divide the x-axis into intervals. We then choose test points within each interval to determine the sign of the function.

- For x < 3, choose x = 2. Substituting x = 2 into the function gives a positive value. - For 3 < x < 5, choose x = 4. Substituting x = 4 into the function gives a negative value. - For x > 5, choose x = 6. Substituting x = 6 into the function gives a positive value.

So, the function is positive for x < 3 and x > 5, and negative for 3 < x < 5.

Intervals of Increase and Decrease

The function increases on the interval (-∞, 3) and decreases on the interval (3, 5), then increases again on the interval (5, ∞).

Range of the Function

The range of the function is all real numbers greater than or equal to -1.

Now, let's plot the graph of the function and mark the key points and intervals we've discussed.