Axis of Symmetry: x=- 2 Maximum Value: 1 Domain: All real numbers Range: y≤ 1
Practice makes perfect
We want to identify the axis of symmetry, maximum or minimum value, domain, and range of the given quadratic function. To do so, we will first express it in vertex form, y=a(x-h)^2+k, where a, h, and k are either positive or negative constants.
y=- 3(x+2)^2+1
⇕
y=- 3(x-(- 2))^2+1
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=& - 3(x-( - 2))^2+ 1
We can see that a= - 3, h= - 2, and k= 1. Note that from the vertex form it follows that the coordinates of the vertex are ( -2, 1).
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= - 2. Therefore, the axis of symmetry is the line x= - 2.
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= - 3, which is less than 0. Thus, the parabola opens downwards and we will have a maximum value. The minimum or maximum value of a parabola is always the y-coordinate of the vertex, k. For this function, it is k= 1.
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 maximum value of the function is 1, the range is all real numbers less than or equal to 1.
Range:& y≤ 1
