I don't understand how to find the maximum and minimum.
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point.
15.f(x) = -x2 + 2x - 4
20.f(x) = 2x2 - 4x
19.f(x) = x2 - 2x + 1
30.f(x) = -3x2 - 3x
#15 and #30 open downwards, so they have a maximum
#20 and #19 open upwards, so they have a minimum.
change the equations to the form
f(x) = a(x-h)^2 + k by completing the square
You should have learned how to do this, and read the vertex from that form.
what are the maximum value and minimum value of (x)=12sin(2x-2pi/3)-5l+3
2x2- 16x+4= i do not know
Find the minimum and maximum possible areas of a rectangle measuring 2 km by 5 km.
To determine whether a quadratic function has a minimum or maximum value, we need to look at the leading coefficient of the function.
If the leading coefficient (the coefficient of the x^2 term) is positive, the quadratic function will have a minimum value.
If the leading coefficient is negative, the quadratic function will have a maximum value.
Now let's find the coordinates of the minimum or maximum point for each of the given quadratic functions.
1. f(x) = -x^2 + 2x - 4:
The leading coefficient is -1, so this quadratic function has a maximum value. To find the coordinates of the maximum point, we can use the vertex formula.
The x-coordinate of the maximum point can be found using the formula x = -b / (2a), where a = -1 and b = 2.
Plugging in the values, we get x = -2 / (2*(-1)) = -2 / -2 = 1.
To find the y-coordinate, we substitute the x-coordinate into the function: f(1) = -(1)^2 + 2(1) - 4 = -1 + 2 - 4 = -3.
Therefore, the maximum point is (1, -3).
2. f(x) = 2x^2 - 4x:
The leading coefficient is 2, so this quadratic function has a minimum value. Again, we can use the vertex formula to find the coordinates of the minimum point.
The x-coordinate of the minimum point can be found using the formula x = -b / (2a), where a = 2 and b = -4.
Plugging in the values, we get x = -(-4) / (2*2) = 4 / 4 = 1.
To find the y-coordinate, we substitute the x-coordinate into the function: f(1) = 2(1)^2 - 4(1) = 2 - 4 = -2.
Therefore, the minimum point is (1, -2).
3. f(x) = x^2 - 2x + 1:
The leading coefficient is 1, which is positive, so this quadratic function has a minimum value.
To find the coordinates of the minimum point, we can again use the vertex formula.
The x-coordinate of the minimum point can be found using the formula x = -b / (2a), where a = 1 and b = -2.
Plugging in the values, we get x = -(-2) / (2*1) = 2 / 2 = 1.
To find the y-coordinate, we substitute the x-coordinate into the function: f(1) = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0.
Therefore, the minimum point is (1, 0).
4. f(x) = -3x^2 - 3x:
The leading coefficient is -3, so this quadratic function has a maximum value. We can use the vertex formula to find the coordinates of the maximum point.
The x-coordinate of the maximum point can be found using the formula x = -b / (2a), where a = -3 and b = -3.
Plugging in the values, we get x = -(-3) / (2*(-3)) = 3 / (-6) = -1/2.
To find the y-coordinate, we substitute the x-coordinate into the function: f(-1/2) = -3(-1/2)^2 - 3(-1/2) = -3(1/4) + 3/2 = -3/4 + 6/4 = 3/4.
Therefore, the maximum point is (-1/2, 3/4).
So, for the given quadratic functions:
- f(x) = -x^2 + 2x - 4 has a maximum point at (1, -3).
- f(x) = 2x^2 - 4x has a minimum point at (1, -2).
- f(x) = x^2 - 2x + 1 has a minimum point at (1, 0).
- f(x) = -3x^2 - 3x has a maximum point at (-1/2, 3/4).