a Begin by writing the function in the vertex form. Then, determine its vertex.
B
b Begin by writing the function in the vertex form. Then, determine its vertex.
C
c Begin by writing the function in the vertex form. Then, determine its vertex.
D
d Begin by writing the function in the vertex form. Then, determine its vertex.
E
e Begin by writing the function in the vertex form. Then, determine its vertex.
F
f Begin by writing the function in the vertex form. Then, determine its vertex.
A
aSymmetry: It is symmetric about x=1.
Verification: See solution.
B
bSymmetry: It is symmetric about x=-1.
Verification: See solution.
C
cSymmetry: It is symmetric about x=3.
Verification: See solution.
D
dSymmetry: It is symmetric about x=-2.
Verification: See solution.
E
eSymmetry: It is symmetric about the y-axis.
Verification: See solution.
F
fSymmetry: It is symmetric about x=5.
Verification: See solution.
Practice makes perfect
a We will describe the symmetry of the graph given by the following quadratic function.
f(x)=-(x-1)^2+4
Let's first rewrite it in the vertex form. Remember that the vertex form of a quadratic function is f(x)=a(x-h)^2+ k, where a≠0 and (h, k) is the vertex. With this form, we can conclude that graph of f(x) is symmetric about x=h.
f(x)=-(x-1)^2+4
⇕
f(x)=-1(x-1)^2+ 4
The vertex of the given function is (1,4). Therefore, it is symmetric about x=1. To verify our answer we will use a graphing calculator. To draw a graph on a calculator, we first press the Y= button and type the function in one of the rows. Having written the function, we can push GRAPH to draw it.
As we can see, the graph is symmetric about x=1.
Therefore, our answer is correct.
b Let's first identify the vertex of the function by rewriting it in the vertex form.
f(x)=(x+1)^2-2
⇕
f(x)=1(x-(-1))^2+( -2)Its vertex is (-1,-2), so its graph is symmetric about x=-1. Now we will check whether our answer is correct.
Looking at the graph, we can say that our answer is correct.
c Notice that the given quadratic function is already in vertex form.
f(x)=(x+1)^2-2
⇕
f(x)=2(x-3)^2+ 1Therefore, we can immediately say that its graph is symmetric about x=3. Let's see it on a graphing calculator.
As we can see, it is indeed symmetric about x=3.
d For the next function, we will rewrite it in the vertex form to describe its symmetry.
f(x)=1/2(x+2)^2
⇕
f(x)=1/2(x-(-2))^2+ 0The vertex is (-2,0), and therefore the graph of the function is symmetric about x=-2.
By the above graph we can tell that our answer is correct.
e Let's examine the next function!
f(x)=-2x^2+3
⇕
f(x)=-2(x- )^2+ 3
Since h=0, the graph is symmetric about the y-axis. We can see this on a graphing calculator.
f Finally, we will describe the symmetry of the last graph. Since it's function is already in the vertex form, we can immediately determine its vertex.
f(x)=3(x-5)^2+ 2The vertex of the function is (5,2), so we can conclude that it is symmetric about x=5. Let's check it!
