a Look for the slope and the y-intercept of both functions.
B
b Look for the slope and the y-intercept of g(x). Then, build the equation. Compare it with the equation of f(x).
C
c Notice that the y-intercepts are the same, but the slopes are different.
D
d Look for the slope and the y-intercept of both functions.
E
e Look for the slope and the y-intercept of g(x). Then, build its equation. Compare it with the equation of f(x).
F
f Look for the slope and the y-intercept of g(x). Then, build its equation. Compare it with the equation of f(x).
A
a To obtain g, we must translate the graph of f 5 units up.
B
b To obtain g, we must shrink the graph of f vertically by a factor of 12.
C
c To obtain g, we must shrink the graph of f horizontally.
D
d To obtain g, we must translate the graph of f 8 units down.
E
e To obtain g, we must reflect the graph of f across the x-axis.
F
f To obtain g, we must reflect the graph of f across the y-axis.
Practice makes perfect
a Let's start by analyzing the parent function, f(x)=2x-4.
Next, we will look for the slope and the y-intercept in the given graph.
We can see that the slope of this graph is 2 and its y-intercept is 1.
Function
Slope
y-intercept
f(x)
2
-4
g(x)
2
1
The function g(x) has the same slope as f(x), but their y-intercepts are different. To obtain g we must translate the graph of f 5 units up.
b Again, we first highlight the characteristics of the parent function f(x)=2x-4.
Let's look for the slope and the y-intercept in the given graph.
We can see that the slope of this graph is 1 and its y-intercept is -2. This means its equation is given by g(x)=x-2. Using this equation, we can set a relation between f(x) and g(x).
1/2f(x) = 1/2(2x-4) = x-2 = g(x)
To obtain g we must shrink the graph of f vertically by a factor of 12.
c We begin by pointing out the characteristics of the parent function, f(x)=2x-4.
Now, we look for the slope and the y-intercept in the given graph.
We can see that the y-intercept of of this graph is -4, but we cannot determine the slope of it. However we see that it must be greater than 2, since it is more vertical than the graph of f(x).
g(x) = ax - 4 = f( ax), a > 2
This means that to obtain g we must shrink the graph of f horizontally.
d We begin by looking at the equation of the parent function, f(x)=2x-4.
Let's also study the graph of the function g(x).
In the graph above, we see that the rise is 4 when the run is 2. This implies that the slope is m= 21=2. Additionally, the y-intercept is -12. We can now compare this data with that from the parent function.
Function
Slope
y-intercept
f(x)
2
-4
g(x)
2
-12
The function g(x) has the same slope as f(x), but their y-intercepts are different. To obtain g we must translate the graph of f 8 units down.
e As we have done before, we will start by analyzing the parent function f(x)=2x-4.
Now, let's take a look at the graph of the function g(x).
As we can see, for every unit we move to the right, points on the line move two units down. This means that the slope is -2. Additionally, the y-intercept is 4. By using the slope-intercept form we can find the equation of this line.
g(x)=-2x+4
Additionally, we can state the following relation between the functions f(x) and g(x).
- f(x) = -(2x-4) = -2x+4 = g(x)
To obtain g we must reflect the graph of f across the x-axis.
f We begin by looking at the equation of the parent function, f(x)=2x-4.
Next, we will take a look at the graph of the function g(x).
As we can see, for every unit we move to the right, points on the line move two units down. This means that the slope is -2. Additionally, the y-intercept is -4. By using the slope-intercept form we find the equation of this line.
g(x)=-2x-4
We can state the following relation between the functions f(x) and g(x).
f(- x) = 2(- x) - 4 = -2x - 4 = g(x)
Therefore, to obtain g we must reflect the graph of f across the y-axis.
