Question

Постройте график функции у=2(х-2)²-3. Найдите"нули функции"​

Answer 1

Ответ:

y = 2(x - 2)² - 3.

Находим нули функции:

2(x - 2)² - 3 = 0

2(x² - 4x + 4) - 3 = 0

2x² - 8x + 8 - 3 = 0

2x² - 8x + 5 = 0

D = b² - 4ac. D = 64 - 40 = 24

$x_{1} = \dfrac{-b-\sqrt{D}}{2a} = \dfrac{8-2\sqrt{6}}{4} = \dfrac{4-\sqrt{6}}{2} \\x_{2} = \dfrac{-b+\sqrt{D}}{2a} = \dfrac{8+2\sqrt{6}}{4} = \dfrac{4+\sqrt{6}}{2}$

Для построения графика функции необходимо найти вершину. Найдём её по формулам:

y = 2x² - 8x + 5
$x_{B} = \dfrac{-b}{2a} = \dfrac{8}{4} = 2$

$y_{B} = f(x_{B}) = 2 * 2^{2} - 8 * 2 +5 = 2 * 4 - 16 + 5 = -3$

Вершина параболы: (2; -3). Ось симметрии: х = 2. Ветви направлены вверх. Точка пересечения с осью OY: (0; 5), и можно построить точку (4; 5). Примерные точки пересечения с осью ОХ мы нашли (это нули функции). Строим график.

Answer 2

Graph of the Function y=2(x-2)²-3

To graph the function y=2(x-2)²-3, we can start by identifying the key components of the function and then plotting the graph based on this information.

The function y=2(x-2)²-3 is a quadratic function in the form of y=a(x-h)²+k, where a determines the direction of the parabola, and (h, k) represents the vertex of the parabola.

Key Components of the Function:

- a = 2: This indicates that the parabola opens upwards. - (h, k) = (2, -3): This represents the vertex of the parabola.

Graphing the Function:

Using the information above, we can plot the graph of the function y=2(x-2)²-3. The vertex of the parabola is at (2, -3), and the parabola opens upwards due to the coefficient a=2.

Now, let's find the "zeros of the function" or the x-intercepts.

Finding the Zeros of the Function:

The zeros of the function can be found by setting y=0 and solving for x.

1. Setting y=0: - 0=2(x-2)²-3

2. Solving for x: - 2(x-2)²-3 = 0 - 2(x-2)² = 3 - (x-2)² = 3/2 - x-2 = ±√(3/2) - x = 2 ± √(3/2)

So, the zeros of the function are x = 2 + √(3/2) and x = 2 - √(3/2).

Therefore, the graph of the function y=2(x-2)²-3 will intersect the x-axis at the points (2 + √(3/2), 0) and (2 - √(3/2), 0).

This information can be used to accurately plot the graph of the function and identify the x-intercepts.