y= -2x^2 + 3x - 9
The given equation is a quadratic function in the form of y = ax^2 + bx + c.
Comparing the given equation with the standard form, we can see that a = -2, b = 3, and c = -9.
To find the vertex of the parabola, we can use the vertex formula, which is x = -b / (2a).
Substituting the values a = -2 and b = 3 into the formula, we get x = -3 / (2 * -2) = -3 / -4 = 3/4.
To find the y-coordinate of the vertex, substitute the x-coordinate (3/4) back into the given equation:
y = -2(3/4)^2 + 3(3/4) - 9
y = -2(9/16) + 9/4 - 9
y = -18/16 + 36/16 - 144/16
y = -126/16
y = -63/8
Therefore, the vertex of the parabola is (3/4, -63/8).
The graph of a parabola passes through the points (-4,8) and (2,8). The equation of the axis of symmetry of the graph of the quadratic function is
To find the equation of the axis of symmetry of a parabola, we need to find the x-coordinate of the vertex. The x-coordinate of the vertex is the average of the x-values of the two given points (-4,8) and (2,8), since the axis of symmetry passes through the vertex and is equidistant from the two points.
(x₁ + x₂) / 2 = (-4 + 2) / 2 = -2 / 2 = -1
Therefore, the x-coordinate of the vertex is -1.
The equation of the axis of symmetry is x = -1.
The x-intercepts of the parabola represented by the function are
f(x) = 3x^2 - 9x - 30
To find the x-intercepts of the parabola represented by the function f(x) = 3x^2 - 9x - 30, we need to set the function equal to zero and solve for x.
So, we have:
3x^2 - 9x - 30 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's try factoring first.
We need to find two numbers that multiply to -90 (the product of the coefficient of x^2 and the constant term) and add up to -9 (the coefficient of x). The numbers that satisfy these conditions are -15 and 6.
So, we can rewrite the equation as:
3x^2 - 15x + 6x - 30 = 0
Now, let's factor by grouping:
(3x^2 - 15x) + (6x - 30) = 0
3x(x - 5) + 6(x - 5) = 0
(3x + 6)(x - 5) = 0
Now, we can set each factor equal to zero and solve for x:
3x + 6 = 0 or x - 5 = 0
For the first equation:
3x + 6 = 0
3x = -6
x = -6/3
x = -2
For the second equation:
x - 5 = 0
x = 5
Therefore, the x-intercepts of the parabola are x = -2 and x = 5.
Determine the vertex of the parabola represented by the function
y = -8(x+3)^2 + 5
The given equation is already in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Comparing the given equation with the vertex form, we can see that h = -3 and k = 5.
Therefore, the vertex of the parabola is (-3, 5).
Determine the standard form of the function f(x) = -2(x-3)^2 + 4
To convert the given function f(x) = -2(x-3)^2 + 4 into standard form, we need to expand and simplify the equation.
Let's start by expanding the square term (x-3)^2:
f(x) = -2(x^2 - 6x + 9) + 4
Next, distribute the -2 to each term inside the parentheses:
f(x) = -2x^2 + 12x - 18 + 4
Combine like terms:
f(x) = -2x^2 + 12x - 14
Therefore, the standard form of the function f(x) = -2(x-3)^2 + 4 is f(x) = -2x^2 + 12x - 14.
An explosives company specializes in bringing down old buildings. The time it takes for a particular piece of flying debris to fall to the ground can be modelled by the function h(t) = -18t^2 + 72t + 378, where h is the height of the debris, in metres, and t is the time, in seconds.
How long will it take for the piece of debris to hit the ground after an explosion?