what is the vertex and line symmetry for f(x)=1/2(x+8)^2+4 and what is the max/min value of f(x)?
Is the answer for the vertex (8,4)
is the min value of f(x) (-8,4)
The answer for the vertex is correct, but the value of the minimum is not.
Also, you have not yet found the line of symmetry.
The standard form of a quadratic is
f(x)=a(x-h)²+k
where (h,k) is the vertex.
If a>0, the parabola is concave upwards, so the minimum is the vertex.
If a<0, the parabola is concave downwards, and the maximum is the vertex.
The line of symmetry is a vertical line through the vertex, namely, x=k.
so the ine symmetry would be 8 and the min or max would be 4
so the ine symmetry would be 8 and the min or max would be 4
Actually, I made a mistake of the sign in the above response.
The equation is written as:
f(x)=a(x-h)²+k
so for
f(x)=1/2(x+8)^2+4
h=-8, and k=4
Therefore the vertex is at (h,k) = (-8,4), and the line of symmetry is at x=-8.
The minimum is at the vertex, namely (-8,4) as you suggested.
To find the vertex and line symmetry for the given function f(x) = 1/2(x+8)^2 + 4, we can start by looking at the standard form of a quadratic function: f(x) = a(x-h)^2 + k. In this case, a = 1/2, h = -8, and k = 4.
The vertex of a quadratic function can be determined using the formula (-h, k). So in this case, the vertex would be (-(-8), 4), which simplifies to (8, 4). Therefore, your answer is correct. The vertex for this function is indeed (8, 4).
To determine the line symmetry of a quadratic function, we consider the value of h. The line of symmetry is given by the equation x = h. In this case, x = -8, so the line of symmetry is x = -8.
Now, let's find the maximum or minimum value of f(x). Since the coefficient of the quadratic term (x^2) is positive (1/2 > 0), the parabola opens upwards, which means the vertex represents the minimum point of the graph. Therefore, f(x) = 1/2(x+8)^2 + 4 has a minimum value at the vertex (8, 4).
In conclusion:
- The vertex of f(x) = 1/2(x+8)^2 + 4 is (8, 4).
- The line of symmetry is x = -8.
- The minimum value of f(x) is indeed (8, 4), not (-8, 4).