What is the rate of change of y with respect to x?
Understand the Problem
The question is asking for the rate of change of y with respect to x, which corresponds to the slope of the given line in the graph. To find the slope, we need to determine the change in y over the change in x between two points on the line.
Answer
The rate of change of \( y \) with respect to \( x \) is \( \frac{1}{4} \).
Answer for screen readers
The rate of change of ( y ) with respect to ( x ) is ( \frac{1}{4} ).
Steps to Solve
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Identify Two Points on the Line Choose two clear points on the line shown in the graph. For example, we can identify the points (0, 0) and (4, 1).
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Calculate the Change in y (Δy) Subtract the y-coordinates of the two points: $$ \Delta y = y_2 - y_1 $$ Using our points: $$ \Delta y = 1 - 0 = 1 $$
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Calculate the Change in x (Δx) Subtract the x-coordinates of the two points: $$ \Delta x = x_2 - x_1 $$ Using our points: $$ \Delta x = 4 - 0 = 4 $$
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Calculate the Slope (Rate of Change of y with respect to x) The slope (m) is given by the formula: $$ m = \frac{\Delta y}{\Delta x} $$ Substituting the values we found: $$ m = \frac{1}{4} $$
The rate of change of ( y ) with respect to ( x ) is ( \frac{1}{4} ).
More Information
The slope ( \frac{1}{4} ) indicates that for every increase of 1 unit in ( x ), ( y ) increases by ( \frac{1}{4} ) units. This means the line is positively sloped, showing a direct relationship between ( x ) and ( y ).
Tips
- Confusing the order of ( x ) and ( y ) when calculating changes; always ensure you calculate ( \Delta y ) and ( \Delta x ) correctly.
- Not simplifying the fraction correctly after calculating the slope.
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