I have been working on these 2 problems and I think I am confusing myself. I have never tried this site before but it was recommended to me. I appreciate whoever can help me. Thanks in advance
The first question is
f(x)=-2x^2 + 2x + 8
I need to find the x and y coordinates of the vertex the line of symmetry and the max/min of f(x)
also the value of f(1/2) =17/2
The 2nd question has a fraction and I think this is confusing me on this one
f(x)=1/5(x+7)^2+7
I need to find the vertex line of symmetry max/min and is the value f(-7)=7 a minimum or a maximum?
-2x^2 + 2x + 8 = y
x^2 - x - 4 = -y/2
x^2 - x = -(1/2) y + 4
x^2 - x + 1/4 = -(1/2) y + 4 1/4
(x-1/2)^2 = -(2/4)y - 17/4
(x-1/2)^2 = -(2/4)(y - 8.5)
so
symmetric about the line x = 1/2
and the vertex is on that vertical line
at y = 8.5
so
vertex point = (1/2, 8.5)
LOL, we already know f(1/2) is 17/2 = 8.5
The parabola is upside down (sheds water , so the vertex is at the maximum.
Check that:
If x = 1/2 then
y = -2 (1/2)^2 +2(1/2) + 8
= -1/2 +1+8 = 8.5 check
can you help with the 2nd question? I have stressed myself over this and I am frustrated. I appreciate you help so much
I'd be happy to help you with both of these problems! Let's start with the first question and find the vertex, line of symmetry, and the maximum/minimum of the function f(x) = -2x^2 + 2x + 8.
To find the vertex of a quadratic function in the form f(x) = ax^2 + bx + c, we can use the formula x = -b/2a. In this case, a = -2 and b = 2.
1. Finding the x-coordinate of the vertex:
x = -b/2a = -2/(2*(-2)) = 2/(-4) = -1/2
So, the x-coordinate of the vertex is -1/2.
2. Finding the y-coordinate of the vertex:
To find the y-coordinate, substitute the x-coordinate (-1/2) back into the original function:
f(-1/2) = -2(-1/2)^2 + 2(-1/2) + 8
= -2(1/4) - 1 + 8
= -1/2 - 1 + 8
= -1/2 - 2/2 + 16/2
= -3/2 + 16/2
= 13/2
So, the y-coordinate of the vertex is 13/2.
3. Finding the line of symmetry:
Since the line of symmetry is the vertical line passing through the vertex, the equation of the line of symmetry is x = -1/2.
4. Finding the maximum/minimum:
In this case, since the coefficient of x^2 is negative (-2), the parabola opens downwards, which means the vertex represents the maximum point. Therefore, the function f(x) has a maximum value at the vertex, and the maximum value is 13/2.
Now let's move on to the second question: f(x) = 1/5(x + 7)^2 + 7.
1. Finding the vertex:
Comparing this function to the general form f(x) = a(x - h)^2 + k, we can see that h = -7 and k = 7. So, the vertex is (-7, 7).
2. Finding the line of symmetry:
The line of symmetry is the vertical line passing through the vertex. In this case, it is x = -7.
3. Finding the maximum/minimum:
Since the coefficient of (x + 7)^2 is positive (1/5), the parabola opens upwards, indicating that the vertex represents the minimum point. Therefore, the function f(x) has a minimum value at the vertex, which is 7.
4. Evaluating f(-7):
Substitute x = -7 into the original function:
f(-7) = 1/5(-7 + 7)^2 + 7
= 1/5(0)^2 + 7
= 1/5(0) + 7
= 0 + 7
= 7
The value of f(-7) is 7, which matches the y-coordinate of the vertex.
To summarize:
For the first question, the vertex is (-1/2, 13/2), the line of symmetry is x = -1/2, and the maximum value is 13/2. Additionally, f(1/2) = 17/2.
For the second question, the vertex is (-7, 7), the line of symmetry is x = -7, and the function has a minimum value of 7. Also, f(-7) = 7.
I hope this helps! Let me know if you have any further questions.