The domain of a quadratic is all real numbers.
The range varies based on the function in question.
the vertex is at (-b/2a, f(-b/2a)). If a is positive this is the minimum, if a is negative, this is the maximum
(will tell you the range)
y=x^2+4x-2
the vertex is at (-2,-6). the axis of symmetry goes through the vertex so x=-2
the y-intercept is when x=0 so f(0)=-2, (0,-2)
the range is {y|y is an element of real numbers, y≥-6}
because it opens upward it decreases to the left of the vertex (-∞,-2)
and increase to the right of the vertex (-2,∞)
the x-intercepts are when the function = 0
you can factor, complete the square, or use the quadratic equation
x^2+4x+4-2-4=0
(x+2)^2-6=0
(x+2)^2=6
x+2=±√6
x=-2-√6 or x=-2+√6
the intercepts are (-2-√6,0),(-2+√6,0)
y=-x^2+x-7
for this one I'm going to start by completing the square because it is easier to extract most of the information (should have done this for the last one too)
y=-x^2+x-7
y=-(x^2-x+ )-7
half of -1 is -1/2, squared is 1/4. since we are adding 1/4 inside the parentheses, times -1=-1./4 we are subtracting 1/4. To keep the equation the same we need to add 1/4
y=-(x^2-x+1/4)-7+1/4
y=-(x-1/2)^2-27/4
in this form: y=a(x-h)^2+k
the vertex is (h,k) and a is the vertical stretch/direction. If a is positive it opens up, negative it opens down
the vertex is (1/2,-27/4) and it opens down so that is the maximum
the domain is all real numbers
the range is {y|y is an element of real numbers, y≤-27/4}
the axis of symmetry is x=1/2
the y intercept is (0,-7)
because this opens downward, it increases to the left of the vertex (-∞,1/2)
and decreases to the right (1/2,∞)
because the vertex is below the x-axis and opens down, there are no x-intercepts. If you solve the quadratic equation, you will get a complex solution