Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point.f(x) = x2 + 2x - 9
since the coefficient of x^2 is positive, we know the parabola opens upward, so it minimum.
The vertex of the parabola is at x = -1, so plug that in to get y.
Or, complete the square:
y = x^2+2x+1 - 8
y = (x+1)^2 - 8
y+8 = (x+1)^2
Now you can just read off the coordinates.
Well, let's find out if this quadratic function is a "laughing matter" or a "serious business". To determine whether it has a minimum or maximum value, we can look at the coefficient of the x^2 term. Since the coefficient is positive (+1), we know this quadratic opens upward, which means it has a minimum value.
To find the coordinates of this minimum point, we need to first determine the x-coordinate. We can do this by using the formula x = -b / (2a), where in this case, a = 1 and b = 2.
Plugging in the values, we get x = -2 / (2 * 1) = -1.
Now, let's find the y-coordinate of the minimum point. We can do this by substituting the x-coordinate (-1) back into the function.
f(-1) = (-1)^2 + 2(-1) - 9
= 1 - 2 - 9
= -10
So, the coordinates of the minimum point are (-1, -10). Ta-da!
To determine whether the given quadratic function has a minimum or maximum value, we can look at the coefficient of the quadratic term. In this case, the coefficient of x^2 is 1, which is positive.
In general, if the coefficient of x^2 is positive, the quadratic function opens upwards and has a minimum value. If the coefficient is negative, it opens downwards and has a maximum value.
Therefore, for the given function f(x) = x^2 + 2x - 9, it has a minimum value.
To find the coordinates of the minimum point, we can find the x-coordinate of the vertex using the formula:
x = -b / (2a)
where a is the coefficient of x^2 and b is the coefficient of x.
In this case, a = 1 and b = 2, so we have:
x = -2 / (2*1) = -2 / 2 = -1
The x-coordinate of the minimum point is -1.
To find the y-coordinate of the minimum point, we substitute the x-coordinate into the function:
f(-1) = (-1)^2 + 2(-1) - 9 = 1 - 2 - 9 = -10
Therefore, the coordinates of the minimum point are (-1, -10).
To determine whether the given quadratic function has a minimum value or maximum value, we can use the fact that the coefficient of the quadratic term (x^2 term) is positive.
In the function f(x) = x^2 + 2x - 9, the coefficient of the x^2 term is 1, which is positive. Since the coefficient is positive, the graph of the quadratic opens upwards, indicating that the function has a minimum value.
To find the coordinates of the minimum point, we can use the vertex formula. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula:
Vertex x-coordinate = -b / (2a)
Vertex y-coordinate = f(vertex x-coordinate)
Let's apply this formula to the given quadratic function:
a = 1, b = 2, and c = -9
Vertex x-coordinate = -2 / (2 * 1) = -1
Vertex y-coordinate = f(-1) = (-1)^2 + 2(-1) - 9 = -1 - 2 - 9 = -12
Therefore, the vertex (minimum point) of the quadratic function f(x) = x^2 + 2x - 9 is (-1, -12).