Example
Match each function with its graph.
a)
f(x)=3x2
b)
f(x)=−3x2
c)
f(x)=21x2
1)
2)
3)
Answer:
Function a) matches graph 2
Function b) matches graph 1
Function c) matches graph 3
Function a) f(x)=3x2 means that inputs are squared and then multiplied by three, so the outputs will be greater than they would have been for f(x)=x2. This results in a parabola that has been squeezed, so graph 2 is the best match for this function.
Function b) f(x)=−3x2 means that inputs are squared and then multiplied by negative three, so the outputs will be farther away from the x-axis than they would have been for f(x)=x2, but negative in value, so graph 1 is the best match for this function.
Function c) f(x)=21x2 means that inputs are squared then multiplied by 21, so the outputs are less than they would be for f(x)=x2. This results in a parabola that has been opened wider thanf(x)=x2. Graph 3 is the best match for this function.
If there is no
Example
Match each of the following functions with its graph.
a)
f(x)=x2+3
b)
f(x)=x2−3
1)
2)
Answer:
Function a) f(x)=x2+3 means square the inputs then add three, so every output will be moved up 3 units. The graph that matches this function best is 2.
Function b) f(x)=x2−3 means square the inputs then subtract three, so every output will be moved down 3 units. The graph that matches this function best is 1.
Changing
Example
Match each of the following functions with its graph.
a)
f(x)=x2+2x
b)
f(x)=x2−2x
a)
b)
Answer:
Find the vertex of function a)
f(x)=x2+2x.
a=1,b=2
x-value:
2a−b=2(1)−2=−1
y-value:
f(2a−b)=(−1)2+2(−1)=1−2=−1.
Vertex = (−1,−1), which means the graph that best fits this function is a)
Find the vertex of function b)
f(x)=x2−2x.
a=1,b=−2
x-value:
2a−b=2(1)2=1
y-value:
f(2a−b)=(1)2−2(1)=1−2=−1.
Vertex = (1,−1), which means the graph that best fits this function is b)
Note that the vertex can change if the value for c changes because the y-value of the vertex is calculated by substituting the x-value into the function. Here is a summary of how the changes to the values for a, b, and, c of a quadratic function can change it is graph.
Example
Graph
f(x)=−2x2+3x–3.
Answer:
Before making a table of values, look at the values of a and c to get a general idea of what the graph should look like.
a=−2, so the graph will open down and be thinner than f(x)=x2.
c=−3, so it will move to intersect the y-axis at (0,−3).
To find the vertex of the parabola, use the formula (2a−b,f(2a−b)). Finding the vertex may make graphing the parabola easier.
Vertex formula=(2a−b,f(2a−b))
x-coordinate of vertex:
2a−b=2(−2)−(3)=−4−3=43
y-coordinate of vertex:
f(2a−b)=f(43)f(43)=−2(43)2+3(43)−3=−2(169)+49−3=16−18+49−3=8−9+818−824=−815
Vertex:
(43,−815)
Use the vertex,
(43,−815), and the properties you described to get a general idea of the shape of the graph. You can create a table of values to verify your graph. Notice that in this table, the
x values increase. The
y values increase and then start to decrease again. This indicates a parabola.
x |
f(x) |
−2 |
−17 |
−1 |
−8 |
0 |
−3 |
1 |
−2 |
2 |
−5 |
Connect the points as best you can using a
smooth curve. Remember that the parabola is two mirror images, so if your points do not have pairs with the same value, you may want to include additional points (such as the ones in blue shown below). Plot points on either side of the vertex.
x=21 and
x=23 are good values to include.
The following video shows another example of plotting a quadratic function using the vertex.
https://youtu.be/leYhH_-3rVo