FIND THE VERTEX THE LINE OF SYMMETRY THE MAX OR MIN VALUE OF THE FOLLOWING
F(X)= -X^2+10X+5
If the U.S. were to have another constitutional convention, would we keep a federal system or change it?
Finding the vertex of a parabola in standard form: f(x)=ax^2 + bx + c uses a formula to find the x-coordinate first:
x=-b/(2a)
x=-10/(2*-1)
x=-10/-2
x=5
To find the y-coordinate, substitute this newly found value of x into the original function.
f(5)=-(5)^2+10(5)+5 (careful with PEMDAS!)
f(5)=-25+50+5
f(5)=30
Therefore, your vertex is at: (5,30)
The Line(Axis) of Symmetry is easy from here. Parabolas given in the form y=x^2 will always be symmetric about a line that is vertical and MUST pass through the vertex. So the equation of the line of symmetry is: x = 5.
The Ymin or Ymax value depends on whether the coefficient of the x^2 term is positive or negative. If its positive, then the parabola opens UPWARD, thus giving you a MIN. If the coefficient is negative, then the parabola opens DOWNWARD, thus giving you a MAX.
So, you have a -x^2, so your parabola opens DOWN, giving you a Ymin. Its specific value is also connected to your vertex. It is equal to the y-coordinate of your vertex. So, its Ymin=30.
Hope this helps!
To find the vertex, line of symmetry, and the maximum or minimum value of the function f(x) = -x^2 + 10x + 5, you can use the technique of completing the square. Here's how:
Step 1: Identify the coefficients of the quadratic equation. In this case, a = -1, b = 10, and c = 5.
Step 2: Find the x-coordinate of the vertex using the formula: x = -b/2a. Plug in the values of a and b: x = -10/(2*(-1)) = 5. This gives you the x-coordinate of the vertex.
Step 3: Substitute the x-coordinate of the vertex into the original equation to find the y-coordinate of the vertex. Plug in x = 5 into f(x): f(5) = -5^2 + 10(5) + 5 = -25 + 50 + 5 = 30. So the vertex is (5, 30).
Step 4: The line of symmetry is the vertical line that passes through the vertex. Since the x-coordinate of the vertex is 5, the equation of the line of symmetry is x = 5.
Step 5: Determine whether the vertex represents a maximum or minimum value. Since the coefficient of x^2 is negative (-1), the parabola opens downwards, meaning the vertex represents the maximum value.
Therefore, the vertex is (5, 30), the line of symmetry is x = 5, and the maximum value of the function is 30.