Find the maximum or minimum value:
-x^2+2x+5
3x^2-4x-2
can someone explain these to me step by step
you could use a TI-83 plus graphing calculator...if that helps
also if the coefficient is negative then it has a minimum value and if it's a positive thenn it has a maximum vlaue
You are probably using "completing the square"
I will do the second one, follow the same procedure to do the first
3x^2-4x-2
= 3[x^2 - (4/3)x + 4/9 - 4/9] - 2
= 3[(x-2/3)^2 - 4/9] - 2
= 3(x-2/3^2 - 4/3 - 6/3
= 3(x-2/3)^2 - 10/3
The minimum value is 10/3 when x = 2/3
To find the maximum or minimum values of a quadratic function, you can either use the vertex formula or the completing the square method. Let's go through each step for each function:
1. -x^2 + 2x + 5:
Step 1: Rewrite the quadratic function in standard form: f(x) = -x^2 + 2x + 5
Step 2: Determine the quadratic coefficient (a), linear coefficient (b), and constant term (c): a = -1, b = 2, c = 5
Step 3: Find the x-coordinate of the vertex using the formula: x = -b / (2a)
x = -2 / (2*(-1)) = 2 / (-2) = -1
Step 4: Substitute the x-coordinate of the vertex into the function to get the y-coordinate: f(-1) = -(-1)^2 + 2(-1) + 5 = -1 - 2 + 5 = 2
Step 5: The vertex of the parabola is (-1, 2). Since the coefficient of x^2 is negative, the quadratic function opens downward. Thus, the vertex represents the maximum point.
2. 3x^2 - 4x - 2:
Step 1: Rewrite the quadratic function in standard form: f(x) = 3x^2 - 4x - 2
Step 2: Determine the quadratic coefficient (a), linear coefficient (b), and constant term (c): a = 3, b = -4, c = -2
Step 3: Find the x-coordinate of the vertex using the formula: x = -b / (2a)
x = -(-4) / (2*3) = 4 / 6 = 2 / 3
Step 4: Substitute the x-coordinate of the vertex into the function to get the y-coordinate: f(2/3) = 3(2/3)^2 - 4(2/3) - 2 = 4/3 - 8/3 - 2 = -6/3 = -2
Step 5: The vertex of the parabola is (2/3, -2). Since the coefficient of x^2 is positive, the quadratic function opens upward. Thus, the vertex represents the minimum point.
So, for the function -x^2 + 2x + 5, the maximum value is 2 at x = -1, and for the function 3x^2 - 4x - 2, the minimum value is -2 at x = 2/3.