I'm having problems with one of the questions on my homework. It has 4 parts. Thanks
9) y = -2(x + 5)^2 + 4
A) Find Vertex
B) Find the Zeros
C) Find the y-intercept(s)
D) This function has a:
a) Minimum b) Maximum
By the way, I think I figured out part A. I have the vertex as -2(-5+5)^2+4=4, so the vertex is (-5, 4). Please confirm, thanks.
surely, giben the vertex form of the parabola, you can read off the vertex?
for the zeros, just solve normally the quadratic
-2(x+5)^2 + 4 = 0
the y-intercept is where x=0
The parabola opens downward, so there is no minimum
The vertex is the maximum, which you have already found.
A) I have the maximum height 47.55 or 47.6
B) I have 5.5 seconds that it will hit the ground
To answer the questions related to the given quadratic function, we need to understand some key concepts. The equation given is in vertex form, which represents a quadratic function's graph as a parabola with the vertex at the point (h, k).
In this case, the equation is: y = -2(x + 5)^2 + 4
A) To find the vertex, we can identify the values of h and k from the equation. The vertex form, y = a(x - h)^2 + k, helps us determine the vertex. Comparing this with the given equation, we can see that h = -5 and k = 4. Therefore, the vertex is at (-5, 4).
B) To find the zeros or x-intercepts, we set y = 0 in the equation and solve for x. In this case, we have:
0 = -2(x + 5)^2 + 4
To solve for x, we isolate the term with x:
2(x + 5)^2 - 4 = 0
Next, we can remove the coefficient (2) by dividing the equation by 2:
(x + 5)^2 - 2 = 0
Now, we take the square root of both sides to eliminate the exponent:
√((x + 5)^2 - 2) = √(0)
Simplifying further, we have:
x + 5 = ±√(2)
To isolate x, we subtract 5 from both sides:
x = -5 ± √(2)
So, the zeros or x-intercepts are given by x = -5 + √(2) and x = -5 - √(2).
C) To find the y-intercept, we set x = 0 in the equation and solve for y. In this case, we have:
y = -2(0 + 5)^2 + 4
y = -2(5)^2 + 4
y = -2(25) + 4
y = -50 + 4
y = -46
Therefore, the y-intercept is at (0, -46).
D) To determine whether the function has a minimum or a maximum, we look at the coefficient of the squared term. In this case, the coefficient is -2, which is negative. A negative coefficient indicates that the parabola opens downward, meaning it has a maximum. Thus, the function has a maximum.
In summary:
A) The vertex is (-5, 4).
B) The zeros are x = -5 + √(2) and x = -5 - √(2).
C) The y-intercept is (0, -46).
D) The function has a maximum.