The equation y = 4x - 2 and the table and graph shown at the right describe three different linear functions. Which function has the greatest rate of change? Which has the least? E... The equation y = 4x - 2 and the table and graph shown at the right describe three different linear functions. Which function has the greatest rate of change? Which has the least? Explain.
Understand the Problem
The question is asking to compare the rates of change of three different linear functions represented by the equation y = 4x - 2 and the table provided. The task is to identify which function has the greatest rate of change and which one has the least, along with an explanation.
Answer
The greatest rate of change is $5$, and the least is $4$.
Answer for screen readers
The function represented by the table has the greatest rate of change at $5$, while the function $y = 4x - 2$ has a rate of change of $4$.
Steps to Solve
- Identify the slope of the given equation
The equation is given as $y = 4x - 2$. The slope, which represents the rate of change, can be determined directly from the equation. Here, the slope $m = 4$.
- Extract the data from the table
The table provides the following points:
- When $x = 1$, $y = 5$
- When $x = 2$, $y = 10$
- When $x = 3$, $y = 15$
- When $x = 4$, $y = 20$
To find the rate of change, we calculate the slope using the formula:
$$ m = \frac{\Delta y}{\Delta x} $$
- Calculate the slope from the table
Using the first two points, $(1, 5)$ and $(2, 10)$:
$$ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 $$
Using the second and third points, $(2, 10)$ and $(3, 15)$:
$$ m = \frac{15 - 10}{3 - 2} = \frac{5}{1} = 5 $$
Using the third and fourth points, $(3, 15)$ and $(4, 20)$:
$$ m = \frac{20 - 15}{4 - 3} = \frac{5}{1} = 5 $$
So, the slope for all points from the table is $5$.
- Compare the rates of change
From our calculations:
- The slope for the equation $y = 4x - 2$ is $4$.
- The slope from the table is $5$.
Thus, the function from the table has a greater rate of change.
The function represented by the table has the greatest rate of change at $5$, while the function $y = 4x - 2$ has a rate of change of $4$.
More Information
The rate of change in a linear function is given by the slope, which indicates how much $y$ changes for a unit change in $x$. The higher the slope, the steeper the line, indicating a greater rate of change.
Tips
- Confusing the slope with the y-intercept: Remember that the slope indicates how steep the function is, whereas the y-intercept shows where the function crosses the y-axis.
- Miscalculating the slope: Make sure to subtract the y-values and x-values correctly.
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