Axis of Symmetry: x=6 Minimum Value: 7 Domain: All real numbers Range: y≥ 7
Practice makes perfect
We want to identify the axis of symmetry, maximum or minimum value, domain, and range of the given quadratic function. Note that the formula is already expressed in vertex form, y=a(x-h)^2+k, where a, h, and k are either positive or negative numbers.
y=1/2(x-6)^2+7
It is important to note that we do not need to graph the parabola to identify the desired information. Let's compare the general formula for the vertex form to our equation.
General Formula:y=& a(x- h)^2 + k
Equation:y=& 1/2(x- 6)^2+ 7
We can see that a= 12, h= 6, and k= 7. Note that from the vertex form it follows that the coordinates of the vertex are ( 6, 7).
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images. This line passes through the vertex and follows a specific format, x= h. For this exercise, we have h= 7. Therefore, the axis of symmetry is the line x= 7.
Maximum or Minimum Value
Before we determine the maximum or minimum, recall that if a>0, the parabola opens upwards. Conversely, if a<0, the parabola opens downwards.
In the given function, we have a= 12, which is greater than 0. Thus, the parabola opens upwards and we will have a minimum value. The minimum or maximum value of a parabola is always the y-coordinate of the vertex, k. For this function, it is k= 7.
Domain and Range
Unless there is a specific restriction given in the context of the problem, the domain of a quadratic function is all real numbers. In this case, there is no restriction on the value of x. Since the minimum value of the function is 7, the range is all real numbers greater than or equal to 7.
Range:& y≥ 7
