Question

Строчнооооооо Функцію задано формулою f(x)=-x2+2x+8 1) побудуйте графік функції f(x) 2) знайдіть суму цілих значень аргумента при яких функція набуває не від’ємних значень

Answer 1

Ответ:

$f(x)=-x^2+2x+8\\\\1)\ \ -x^2+2x+8=0\ \ ,\ \ x_{versh}=-\dfrac{b}{2a}=-\dfrac{2}{-2}=1\\\\y_{versh}=y(1)=-1+2+8=9\\\\C(1;9)\ -\ vershina\\\\f(0)=8\\\\-x^2+2x+8=0\ \ \ \Rightarrow \ \ \ x_1=-2\ ,\ x_2=4\ \ ,\ \ \ A(-2;0)\ ,\ B(4;0)\\\\2)\ \ -x^2+2x+8\geq 0\ \ \ \to \ \ \ x^2-2x-8\leq 0\ \ ,\\\\(x+2)(x-4)\leq 0\ \ \ \to \ \ \ \ x\in [-2\ ;\ 4\ ]\\\\symma\ celux:\ \ -2-1+0+1+2+3+4=7$

Answer 2

Function and Graph:

The given function is f(x) = -x^2 + 2x + 8. To plot the graph of this function, we can use a graphing calculator or software. Here is the graph of the function:


Sum of Non-Negative Values:

To find the sum of the integer values of x for which the function f(x) takes non-negative values, we need to determine the range of x values that satisfy this condition.

In order for f(x) to be non-negative, it must be greater than or equal to zero. So we can set up the inequality:

-f(x) ≥ 0

Substituting the given function, we have:

-(-x^2 + 2x + 8) ≥ 0

Simplifying the inequality:

x^2 - 2x - 8 ≥ 0

To find the values of x that satisfy this inequality, we can factorize the quadratic expression:

(x - 4)(x + 2) ≥ 0

The critical points are x = -2 and x = 4. We can use these points to divide the number line into three intervals: (-∞, -2), (-2, 4), and (4, ∞).

Now, we need to determine the sign of the expression (x - 4)(x + 2) in each interval. We can choose a test point from each interval and substitute it into the expression to determine the sign.

For example, let's choose x = -3 from the interval (-∞, -2):

(-3 - 4)(-3 + 2) = (-7)(-1) = 7

Since the result is positive, we know that the expression (x - 4)(x + 2) is positive in the interval (-∞, -2).

Similarly, we can choose x = 0 from the interval (-2, 4):

(0 - 4)(0 + 2) = (-4)(2) = -8

Since the result is negative, we know that the expression (x - 4)(x + 2) is negative in the interval (-2, 4).

Finally, let's choose x = 5 from the interval (4, ∞):

(5 - 4)(5 + 2) = (1)(7) = 7

Since the result is positive, we know that the expression (x - 4)(x + 2) is positive in the interval (4, ∞).

Now, we can determine the values of x for which the function f(x) takes non-negative values. These values are in the intervals where the expression (x - 4)(x + 2) is greater than or equal to zero.

From our analysis, we can see that the function f(x) takes non-negative values in the intervals (-∞, -2] and [4, ∞). To find the sum of the integer values of x in these intervals, we can list the integers in each interval and calculate their sum:

(-∞, -2]: -2, -1, 0, 1, 2

[4, ∞): 4, 5, 6, 7, ...

The sum of the integer values in these intervals is -2 + -1 + 0 + 1 + 2 + 4 + 5 + 6 + 7 + ...