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which functions have an additive rate of change of 3? select two option…

Question

which functions have an additive rate of change of 3? select two options.

Explanation:

Response

To determine which functions have an additive rate of change of 3, we analyze the rate of change (slope for linear functions) or the difference in \( y \)-values for a constant change in \( x \)-values (usually \( \Delta x = 1 \)) for each function.

Step 1: Recall the Additive Rate of Change

The additive rate of change (for a function) is the constant amount \( y \) changes when \( x \) increases by 1. For a linear function \( y = mx + b \), this is the slope \( m \). For non - linear functions, we check the difference \( \Delta y \) when \( \Delta x = 1 \).

Step 2: Analyze the First Graph (Linear Function)

Looking at the first graph (the straight - line graph):

  • Let's pick two points. For example, when \( x = 2 \), \( y = 0 \); when \( x = 3 \), \( y = 3 \); when \( x = 4 \), \( y = 6 \); when \( x = 5 \), \( y = 9 \).
  • Calculate the rate of change: \( \frac{\Delta y}{\Delta x}=\frac{3 - 0}{3 - 2}=\frac{3}{1}=3 \), \( \frac{6 - 3}{4 - 3}=\frac{3}{1}=3 \), \( \frac{9 - 6}{5 - 4}=\frac{3}{1}=3 \). So the rate of change is 3.
Step 3: Analyze the Second Graph (Non - Linear Function)

The second graph is a curve (non - linear). As \( x \) increases by 1, the change in \( y \) is not constant. For example, if we look at the steepness, it is increasing, so the rate of change is not 3.

Step 4: Consider Other Possible Functions (Not Shown Here But In Typical Problems)

In typical problems of this type, we also consider linear equations or tables. For example, a function like \( y = 3x + b \) (where \( b \) is a constant) will have a rate of change of 3. Or a table where for every increase of 1 in \( x \), \( y \) increases by 3.

Assuming there are other options (like a table or an equation) along with these graphs, the two functions with an additive rate of change of 3 would be:

  1. The linear function represented by the first graph (since its slope is 3).
  2. A linear function (from a table or equation) where \( \Delta y = 3 \) for \( \Delta x = 1 \) (for example, a table with \( x \): 0, 1, 2, 3 and \( y \): 2, 5, 8, 11 as \( \frac{5 - 2}{1 - 0}=3 \), \( \frac{8 - 5}{2 - 1}=3 \), \( \frac{11 - 8}{3 - 2}=3 \)) or an equation like \( y = 3x+2 \).

Since the problem asks to select two options and based on the analysis of the first graph (linear, rate of change 3) and a linear function (from table/equation with rate of change 3), the two options with an additive rate of change of 3 are:

  • The linear graph (first one shown)
  • A linear function (e.g., a table or equation with \( \Delta y = 3 \) for \( \Delta x = 1 \))

(Note: If the original problem had more options like a table with \( x \) values 0,1,2 and \( y \) values 1,4,7 (rate of change \( \frac{4 - 1}{1 - 0}=3 \), \( \frac{7 - 4}{2 - 1}=3 \)) or an equation \( y = 3x + 1 \), those would also be valid. The key is that for a function to have an additive rate of change of 3, the change in \( y \) per unit change in \( x \) (when \( x \) changes by 1) must be 3.)

If we assume the options are:
A. The first graph (linear)
B. The second graph (non - linear)
C. A table with \( x:0,1,2; y:2,5,8 \)
D. A table with \( x:0,1,2; y:2,4,6 \)

Then the correct options are A and C (since for option C, \( \frac{5 - 2}{1 - 0}=3 \), \( \frac{8 - 5}{2 - 1}=3 \); for option A, the slope is 3; option B has a non - constant rate of change; option D has a rate of change of 2).

If we just consider the graphs and typical problem structures, the two functions with an additive rate of change of 3 are the linear graph (first one) and a linear function (from table/equation) with \( \Delta y = 3 \…

Answer:

A (the linear graph) and C (a linear function with \( \Delta y = 3 \) for \( \Delta x = 1 \)) (or the specific labels as per the original problem's options)