a Can the function be negative for any value of x?
B
b Use that the graph is symmetric about a certain value of x.
C
c Can the function be negative for any value of x?
D
d Use that the graph is symmetric about a certain value of x.
E
e Compare the graph left of the lowest or highest point with the graph to the right of the same point.
A
a x=1
B
b x=0
C
c x=- 2
D
d x=0
E
e See solution.
Practice makes perfect
a Notice that the expression in the parentheses of the given function is squared. This means that the function will never be negative for any value of x. Additionally, when x= 1 the function and the expression inside the parentheses will both equal 0.
y=( 1-1)^2 ⇔ y=0
This is the smallest possible output. Let's look at the function.
We can see that the graph is symmetric about x=1. That is also where it has its lowest point.
b This time, let's begin by looking at the graph of the function.
The graph is symmetric about x=0. This is also where it has its lowest point.
c This function contains a pair of parentheses that are squared. This means that the function will never be negative. However, the function can become 0. This will happen when x+2=0, which is when x= - 2.
y=( - 2+2)^2 ⇒ y=0
Let's look at the graph of this function.
The graph is symmetric about x=- 2. That is also where we find the function's smallest possible output.
d Looking at the graph of this function, we can see that the parabola opens downward this time.
Therefore, this function has a largest possible output. The highest point is found by using that the graph is symmetric about x=0. The graph is at its highest point at x=0.
e For each of the functions, the highest or lowest point can be found at the x-value about which the graph is symmetric.
