Notice that each piece of function starts where the previous function ends. Find the y-intercept of each function depending on its starting point.
Function:
f(x)=0.014x if 0≤ x ≤ 100 0.024x-1 if 100< x ≤ 500 0.034x-6 if 500< x
Graph:
Practice makes perfect
We have 3 pieces of information about the earning of a savings account. We will write a piece-wise function that models the earning and we will graph the function piece by piece. Let's start!
First Piece
The first piece of information is the following.
The account earns 1.4 % simple interest
for balances of $100 or less.
In this case, since the interest states the rate of change in the balance, we will consider it as a slope. If x represents the balance and y=f(x) represents the interest paid, we can write the following function. Notice that, the amount of interest is $0 for $0 balance. Therefore, the y-intercept will be 0.
f(x)= 0.014x if 0≤ x ≤ 100
Next, we will find the end points of our function to graph it.
x
Function
Substitution
y=f(x)
0
f(x)=0.014x
f( 0)=0.014* 0
0
100
f(x)=0.014x
f( 100)=0.014* 100
1.4
Now, we can graph the first piece of our function using the endpoints (0,0) and (100,1.4). We will show the endpoints with closed points since they are included in the solution set.
Second Piece
The next information tells us the following.
The account earns 2.4 % simple interest
for balances greater than $100 and up to $500.
Let's write the function. In this case, the y-intercept will be different than 0 because the starting point of the function is (100,1.4).
f(x)= 0.024x+ b if 100< x ≤ 500
We will find b by substituting the point (100,1.4) into the function.
Thus, the function can be written as the following.
f(x)= 0.024x-1 if 100< x ≤ 500
Let's determine its endpoint to draw it.
x
Function
Substitution
y=f(x)
100
f(x)=0.014x
f( 0)=0.014* 100
1.4
500
f(x)=0.024x-1
f( 500)=0.024* 500-1
11
The endpoints are (100,1.4) and (500,11). Let's graph it!
Third Piece
The last piece of information is given below.
The account earns 3.4 % simple interest
for balances greater than $500.
The function models this piece can be written as the following.
f(x)= 0.034x+ b if 500< x
Notice that the starting point of the function is the point (500,11) from the previous part. Let's find the y-intercept of the function by substituting the point into the function.
Thus, the function can be written as the following.
f(x)= 0.034x-6 if 500< x
Since the function starts from the point (500,11) and goes without any upper boundary, we can add its function as the following.
The piecewise function can be written as shown below.
f(x)= 0.014x if 0≤ x ≤ 100 0.024x-1 if 100< x ≤ 500 0.034x-6 if 500< x
