F(X)= -3x^2 -30x-70
what is the vertex
what is the axsis of symmetry
what are the intercept
what is the domain
what is the range
intervals increase
intervals decrease
you can see it's a parabola. By completing the square, you get
y = -3(x^2 + 10x + 25) + 5
y = -3(x+5)^2 + 5
you should be able to read off most of the answers you need from that.
im still having a hard time please help a little more
review what you know about the parabola
y = a(x-h)^2 + k
Here, we have
y = -3(x+5)^2 + 5
vertex at (-5,2)
axis at x = -5
y-intercept at (0,5)
x-intercepts where 3x^2+30x+70 = 0, or
x = 1/3 (-15±√15)
domain is all reals
range is all reals <= 5
increase for x < -5
decrease for x > -5
To find the vertex of the given function f(x) = -3x^2 - 30x - 70, you can use the formula x = -b/2a. In this case, a = -3 and b = -30.
1. Vertex:
To find the x-coordinate of the vertex, use the formula x = -b/2a.
So, x = -(-30)/(2*(-3)) = 30/(-6) = -5.
To find the y-coordinate of the vertex, substitute the x-coordinate (-5) back into the equation f(x) = -3x^2 - 30x - 70.
So, f(-5) = -3(-5)^2 - 30(-5) - 70 = -3(25) + 150 - 70 = -75 + 150 - 70 = 5.
Therefore, the vertex of the function is (-5, 5).
2. Axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. So, the equation of the axis of symmetry is x = -5.
3. Intercepts:
To find the x-intercepts, set f(x) = -3x^2 - 30x - 70 equal to zero and solve for x:
-3x^2 - 30x - 70 = 0.
This quadratic equation doesn't factor easily, so you can use the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a).
In this case, a = -3, b = -30, and c = -70.
Using the quadratic formula: x = (-(-30) ± √((-30)^2 - 4*(-3)*(-70)))/(2*(-3)).
Simplifying: x = (30 ± √(900 - 840))/(-6), x = (30 ± √(60))/(-6).
The discriminant (b^2 - 4ac) is 60, which is greater than zero. So, there are two distinct x-intercepts.
Solving further: x = (30 + √60)/(-6) and x = (30 - √60)/(-6).
To find the y-intercept, simply substitute x = 0 into the equation f(x) = -3x^2 - 30x - 70.
f(0) = -3(0)^2 - 30(0) - 70 = 0 - 0 - 70 = -70.
Therefore, the x-intercepts are approximately x = -1.077 and x = -9.923, and the y-intercept is (0, -70).
4. Domain:
The domain of the function is all possible x-values for which the function is defined.
Since a quadratic function is defined for all real numbers, the domain is (-∞, ∞) or "all real numbers."
5. Range:
The range of the function is all possible y-values that the function can take.
Since the quadratic function has a negative coefficient for the x^2 term, the vertex is the maximum point.
So, the range is (-∞, 5] or "all y-values less than or equal to 5."
6. Intervals of Increase and Decrease:
To find intervals of increase and decrease, you can look at the coefficient of the x^2 term, which is -3 in this case.
Since the coefficient is negative, the parabola opens downward, and the function is decreasing in the entire domain.
Therefore, the intervals of increase are none, and the intervals of decrease are (-∞, ∞).