Question

Па­ра­бо­ла про­хо­дит через точки A(0; – 6), B( – 5; – 1), C(1; – 1). Най­ди­те ко­ор­ди­на­ты её вер­ши­ны.

Answer 1

Відповідь: Р(-2;-10)

Пояснення

Answer 2

Finding the Vertex of a Parabola

To find the coordinates of the vertex of a parabola, we can use the formula:

x = -b / (2a)

where a and b are the coefficients of the quadratic equation in standard form: ax^2 + bx + c = 0.

In this case, we have three points: A(0, -6), B(-5, -1), and C(1, -1). We can use these points to form a system of equations and solve for the coefficients a and b.

Let's start by finding the equation of the parabola using the point A(0, -6) and the vertex formula.

Solving for the Coefficients

Using the point A(0, -6), we can substitute the values into the equation:

-6 = a(0)^2 + b(0) + c

Simplifying, we get:

-6 = c

Now, let's use the points B(-5, -1) and C(1, -1) to form two more equations.

Using the point B(-5, -1):

-1 = a(-5)^2 + b(-5) + c

Simplifying, we get:

25a - 5b + c = -1 Using the point C(1, -1):

-1 = a(1)^2 + b(1) + c

Simplifying, we get:

a + b + c = -1 Now, we have a system of three equations:

-6 = c (Equation 1)

25a - 5b + c = -1 (Equation 2)

a + b + c = -1 (Equation 3)

We can solve this system of equations to find the values of a, b, and c.

Solving the System of Equations

Substituting Equation 1 into Equations 2 and 3, we get:

25a - 5b - 6 = -1 (Equation 4)

a + b - 6 = -1 (Equation 5)

Simplifying Equation 4, we get:

25a - 5b = 5 (Equation 6)

Subtracting Equation 5 from Equation 6, we eliminate a:

24a - 6b = 6 (Equation 7)

Now, let's solve Equations 5 and 7 simultaneously.

Multiplying Equation 5 by 24, we get:

24a + 24b - 144 = -24 (Equation 8)

Subtracting Equation 7 from Equation 8, we eliminate a:

48b - 138 = -30 (Equation 9)

Simplifying Equation 9, we get:

48b = 108 (Equation 10)

Dividing both sides of Equation 10 by 48, we find:

b = 2.25

Now, substituting the value of b into Equation 5, we can solve for a:

a + 2.25 - 6 = -1

Simplifying, we get:

a = 4.75

Now that we have the values of a and b, we can find the x-coordinate of the vertex using the formula:

x = -b / (2a)

Substituting the values, we get:

x = -2.25 / (2 * 4.75)

Simplifying, we find:

x ≈ -0.237

To find the y-coordinate of the vertex, we substitute the x-coordinate back into the equation of the parabola:

y = ax^2 + bx + c

Substituting the values of a, b, and c we found earlier, we get:

y = 4.75x^2 + 2.25x - 6

Substituting x ≈ -0.237, we find:

y ≈ -6.237

Therefore, the coordinates of the vertex of the parabola passing through the points A(0, -6), B(-5, -1), and C(1, -1) are approximately (-0.237, -6.237).

Please note that the calculations were approximated for simplicity.