a Multiply both sides of the given equation by - 1 and simplify.
B
b Calculate ( b2 )^2.
C
c Multiply both sides of the given equation by - 1 and simplify.
D
d Consider the sign of a in the graphing form of a quadratic function, y=a(x-h)^2+k.
A
a - y = x^2-4x+1
B
b - y=(x-2)^2-3
C
c y=- (x-2)^2+3
D
dVertex: (2,3) Maximum Value: 3
Practice makes perfect
a We want to find the maximum value of the given function. To do this we will rewrite the given function in graphing form by completing the square. Let's start by multiplying both sides of the given equation by - 1.
b We want to complete the square on the right-hand side of the function. Recall that in a quadratic expression b is the linear coefficient. For the function we obtained in Part A, we have that b=- 4. Let's now calculate ( b2 )^2.
d Now, we can state the vertex and the maximum value of the parabola. It is important to note that we do not need to graph the parabola to identify the desired information.
Vertex
Let's compare the general formula for the graphing form to our equation.
General Formula:y=& a(x- h )^2 +k
Equation:y=& - 1(x- 2)^2+3
We can see that a= - 1, h= 2, and k=3.
The vertex of a quadratic function written in graphing form is the point ( h,k). For this exercise, we have h= 2 and k=3. Therefore, the vertex of the given equation is ( 2,3).
Maximum 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 our function we have a= - 1, which is less than 0. Thus, as Thao said, 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=3.
