How to find coordinates of vertex, and find out if the function has minimum or maximum value and find that value?
f(x)=2x^2+8x+9
&
f(x)=-x^2-2x+7
Please explain..I'm very lost. :(
you know the vertex of y=a(x-h)^2+k is at (h,k). The minimum value of (x-h) is zero, when x=h. For any other value of x, (x-h)^2 gets larger.
So, you want to get your function in that form and then you can just read off the coordinates of the vertex.
f(x) = 2x^2+8x+9
= 2(x^2+4x+4)+1
= 2(x+2)^2 + 1
So, the vertex is at (-2,1)
f(x) = -x^2-2x+7
= -(x^2+2x+1)+8
= -(x+1)^2+8
So the vertex is at (-1,8) and the parabola opens downward.
Thank you very much Steve.
To find the coordinates of the vertex and determine whether the function has a minimum or maximum value, follow these steps:
1. Start with the general form of a quadratic function: f(x) = ax^2 + bx + c.
2. Identify the values of a, b, and c for each function:
For f(x) = 2x^2 + 8x + 9:
- a = 2
- b = 8
- c = 9
For f(x) = -x^2 - 2x + 7:
- a = -1
- b = -2
- c = 7
3. The x-coordinate of the vertex can be found using the formula: x = -b / (2a).
For f(x) = 2x^2 + 8x + 9:
- Substitute a = 2 and b = 8 into the formula:
x = -8 / (2 * 2) = -2
- The x-coordinate of the vertex for this function is -2.
For f(x) = -x^2 - 2x + 7:
- Substitute a = -1 and b = -2 into the formula:
x = -(-2) / (2 * -1) = 1
- The x-coordinate of the vertex for this function is 1.
4. To find the y-coordinate of the vertex, substitute the x-coordinate (found in step 3) into the original function.
For f(x) = 2x^2 + 8x + 9:
- Substitute x = -2 into the function:
f(-2) = 2(-2)^2 + 8(-2) + 9
= 8 - 16 + 9
= 1
- The y-coordinate of the vertex for this function is 1.
For f(x) = -x^2 - 2x + 7:
- Substitute x = 1 into the function:
f(1) = -(1)^2 - 2(1) + 7
= -1 - 2 + 7
= 4
- The y-coordinate of the vertex for this function is 4.
5. To determine whether the function has a minimum or maximum value, look at the value of a:
- If a > 0, the parabola opens upwards, and the vertex represents a minimum value.
- If a < 0, the parabola opens downwards, and the vertex represents a maximum value.
For f(x) = 2x^2 + 8x + 9:
- Since a = 2 (which is > 0), the function has a minimum value at the vertex.
For f(x) = -x^2 - 2x + 7:
- Since a = -1 (which is < 0), the function has a maximum value at the vertex.
6. Finally, find the minimum or maximum value by substituting the x-coordinate of the vertex into the original function.
For f(x) = 2x^2 + 8x + 9:
- The minimum value is found by substituting x = -2 into the function:
f(-2) = 2(-2)^2 + 8(-2) + 9
= 1
For f(x) = -x^2 - 2x + 7:
- The maximum value is found by substituting x = 1 into the function:
f(1) = -(1)^2 - 2(1) + 7
= 4
Therefore, the function f(x) = 2x^2 + 8x + 9 has a minimum value of 1 located at the vertex (-2, 1), while the function f(x) = -x^2 - 2x + 7 has a maximum value of 4 located at the vertex (1, 4).
To find the coordinates of the vertex and determine if the function has a minimum or maximum value, you can follow these steps:
Step 1: Take the given function in the form f(x) = ax^2 + bx + c.
For the first function, f(x) = 2x^2 + 8x + 9, the values of a, b, and c are a = 2, b = 8, and c = 9.
For the second function, f(x) = -x^2 - 2x + 7, the values of a, b, and c are a = -1, b = -2, and c = 7.
Step 2: The x-coordinate of the vertex can be found using the formula x = -b / (2a). This formula gives the x-value of the vertex, which is the axis of symmetry for the parabolic function.
For the first function, substituting the values of a and b, we have x = -(8) / (2 * 2) = -8 / 4 = -2.
For the second function, substituting the values of a and b, we have x = -(-2) / (2 * -1) = 2 / -2 = -1.
So, for the first function, the x-coordinate of the vertex is -2, and for the second function, the x-coordinate of the vertex is -1.
Step 3: To find the y-coordinate of the vertex, substitute the x-coordinate obtained in Step 2 into the given function.
For the first function, f(-2) = 2*(-2)^2 + 8*(-2) + 9 = 8 - 16 + 9 = 1.
For the second function, f(-1) = -(-1)^2 - 2*(-1) + 7 = -1 + 2 + 7 = 8.
So, for the first function, the y-coordinate of the vertex is 1, and for the second function, the y-coordinate of the vertex is 8.
Step 4: Determine if the parabolic function has a minimum or maximum value.
- If the coefficient 'a' is positive, the parabola opens upwards and has a minimum value at the vertex.
- If the coefficient 'a' is negative, the parabola opens downwards and has a maximum value at the vertex.
For the first function, since a = 2 (positive), the parabola opens upwards and has a minimum value at the vertex.
For the second function, since a = -1 (negative), the parabola opens downwards and has a maximum value at the vertex.
Step 5: Finally, the value of the function at the vertex is the y-coordinate of the vertex.
For the first function, the minimum value is 1.
For the second function, the maximum value is 8.
So, the coordinates of the vertex and the minimum/maximum values for the given functions are as follows:
1. For f(x) = 2x^2 + 8x + 9:
- Vertex: (-2, 1)
- Minimum value: 1
2. For f(x) = -x^2 - 2x + 7:
- Vertex: (-1, 8)
- Maximum value: 8