Which linear function has the greatest initial value? Which has the greatest rate of change?

Question

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Function A line Function B line Function C
x 0 2 4 6
y 0 5 10 15
y = 3x – 1 A graph of a diagonal line on a coordinate plane going up and to the right. The line begins at the point zero comma 2 and passes through the point 3 comma 3.

Function
C
has the greatest initial value.
Function
A
has the greatest rate of change.

Answer 1

To determine the initial value and the rate of change for the functions, we analyze each one:

1. Initial Value: This refers to the y-intercept of the function, which is the value of \(y\) when \(x = 0\).

   • For Function A: From the graph, when \(x = 0\), \(y = 2\).
   • For Function B: The graph shows that when \(x = 0\), \(y = 5\).
   • For Function C: The y-value when \(x = 0\) appears to be \(0\) (from y=3x-1, when \(x = 0\), \(y = -1\)).

   Conclusion: Function B has the greatest initial value.

2. Rate of Change: This is determined by the slope of the line. The slope can be extracted from the linear equation or the change in \(y\) over the change in \(x\) from graphical points.

   • Function A: The equation \(y = 3x - 1\) has a slope of \(3\).
   • Function B: It’s not explicitly provided, but from the y-values, if we glean two points (0, 5) and (2, 10), the slope is (10-5)/(2-0) = 5/2 = 2.5.
   • Function C: Assuming \(y = 3x - 1\), if it continued appropriately, the slope is also 3.

   Conclusion: Function A and Function C share the same rate of change at \(3\), which is greater than Function B’s rate of change. Therefore, both A and C have the greatest rate of change.

Final Answer:

• Function B has the greatest initial value.
• Function A (or C) has the greatest rate of change.

If you need specific dropdown selections:

• Greatest Initial Value: Function B
• Greatest Rate of Change: Function A (or C, depending on how dropdown options are structured).