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?
Hello! I'm here to help you with these questions. Let's start with the first one.
1. For the function f(x) = -2x^2 + 2x + 8, you want to find the x and y coordinates of the vertex, the line of symmetry, and whether it has a maximum or minimum.
To find the x-coordinate of the vertex, you can use the formula x = -b / (2a), where a, b, and c are coefficients from the general quadratic equation form ax^2 + bx + c = 0. In this case, a = -2 and b = 2. So, substituting the values into the formula, we have x = -2 / (2*(-2)) = -2 / (-4) = 1/2. Therefore, the x-coordinate of the vertex is 1/2.
To find the y-coordinate of the vertex, substitute the x-coordinate (-2/4) back into the original equation: f(1/2) = -2(1/2)^2 + 2(1/2) + 8. Simplifying this expression, we get f(1/2) = -1/2 + 1 + 8 = 9 - 1/2 = 17/2. So, the y-coordinate of the vertex is 17/2.
The vertex of the parabola represented by f(x) = -2x^2 + 2x + 8 is therefore (1/2, 17/2).
The line of symmetry for a parabola is given by the equation x = h, where h is the x-coordinate of the vertex. In this case, the line of symmetry is x = 1/2.
Now, to determine whether the parabola has a maximum or a minimum, you can examine the coefficient of the x^2 term. If the coefficient is positive (as it is in this case), the parabola opens downwards and has a maximum. Conversely, if the coefficient is negative, the parabola opens upwards and has a minimum. Therefore, f(x) = -2x^2 + 2x + 8 has a maximum.
Moving on to the second question:
2. For the function f(x) = 1/5(x+7)^2 + 7, you need to find the vertex, the line of symmetry, and determine whether the value f(-7) = 7 is a minimum or maximum.
To find the vertex, you can use the same formula as before, x = -b / (2a). In this case, a = 1/5 and b = 7. Substituting the values into the formula, we have x = -7 / (2*(1/5)) = -7 / (2/5) = -7 * (5/2) = -35/2. The x-coordinate of the vertex is therefore -35/2.
To find the y-coordinate of the vertex, substitute the x-coordinate (-35/2) back into the original equation: f(-35/2) = (1/5)(-35/2 + 7)^2 + 7. Simplifying this expression step by step, we have f(-35/2) = (1/5)(-35/2 - 14/2)^2 + 7 = (1/5)(-49/2)^2 + 7 = (1/5)(2401/4) + 7 = 1201/20 + 7.
To add the fractions, you need a common denominator, which is 20. So, 1201/20 + 7 = 1201/20 + 7*20/20 = 1201/20 + 140/20 = (1201 + 140)/20 = 1341/20. Therefore, the y-coordinate of the vertex is 1341/20.
The vertex of the parabola represented by f(x) = 1/5(x+7)^2 + 7 is (-35/2, 1341/20).
The line of symmetry, just like in the previous question, is x = h, where h is the x-coordinate of the vertex. So, in this case, the line of symmetry is x = -35/2.
Now, to determine whether the value f(-7) = 7 is a minimum or maximum, we can compare it to the y-coordinate of the vertex. The y-coordinate of the vertex is 1341/20, which is greater than 7. Hence, f(-7) = 7 represents a minimum.
I hope this explanation helps you understand how to solve these problems! If you have any further questions, feel free to ask.