Question

the equation $y = 4x + 60$ and the table each describe a linear function. compare the properties of the functions.\
\

x	10	20	30	40	\
y	60	80	100	120	\

\
select all that apply.\
\
a. the linear function described by the equation has the greater rate of change.\
\
b. the linear function described by the table has the greater rate of change.\
\
c. the linear function described by the equation has the greater initial value.\
\
d. the initial values are equal.\
\
e. the linear function described by the table has the greater initial value.\
\
f. the rates of change are equal.

Explanation:

Step1: Find rate of change for equation

The equation is \( y = 4x + 60 \), which is in slope - intercept form \( y=mx + b \), where \( m \) is the rate of change (slope). So the rate of change for the equation is \( m_1 = 4 \).

Step2: Find rate of change for table

For a linear function represented by a table, the rate of change (slope) is calculated using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points from the table, say \( (x_1,y_1)=(10,60) \) and \( (x_2,y_2)=(20,80) \). Then \( m_2=\frac{80 - 60}{20 - 10}=\frac{20}{10} = 2 \). We can check with other points, for example, \( (20,80) \) and \( (30,100) \): \( m_2=\frac{100 - 80}{30 - 20}=\frac{20}{10}=2 \), and \( (30,100) \) and \( (40,120) \): \( m_2=\frac{120 - 100}{40 - 30}=\frac{20}{10}=2 \). So the rate of change for the table is \( m_2 = 2 \).

Step3: Compare rates of change

Since \( 4>2 \), the rate of change of the function described by the equation is greater than that of the function described by the table.

Step4: Find initial value for equation

The initial value (y - intercept) of the equation \( y = 4x+60 \) is \( b_1 = 60 \) (when \( x = 0 \), \( y=60 \)).

Step5: Find initial value for table

To find the initial value (y - intercept) of the function in the table, we can use the slope - intercept form \( y=mx + b \). We know \( m = 2 \) and we can use a point, say \( (10,60) \). Substitute into \( y=mx + b \): \( 60=2\times10 + b \), so \( 60 = 20 + b \), then \( b=60 - 20=40 \). Let's verify with another point, \( (20,80) \): \( 80=2\times20 + b\Rightarrow80 = 40 + b\Rightarrow b = 40 \). So the initial value for the table is \( b_2=40 \).

Step6: Compare initial values

Since \( 60>40 \), the initial value of the function described by the equation is greater than that of the function described by the table.

Answer:

A. The linear function described by the equation has the greater rate of change.
C. The linear function described by the equation has the greater initial value.