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The rate of change in linear and quadratic functions helps us understand how the output of these functions changes in relation to changes in their input values. Linear functions have a constant rate of change, which is the slope of the line. Quadratic functions have a variable rate of change, showing acceleration or deceleration in the change of the output values.
Function: y = 2x + 3
This linear function has a constant rate of change (slope) of 2. This means for every one unit increase in x
, y
increases by 2 units.
Function: y = 4x + 1
The rate of change (slope) for this linear function is 4. For every one unit increase in x
, y
decreases by 4 units, indicating a downward slope.
Function: y = x^2
The rate of change for this quadratic function varies with x
. As x
increases or decreases from 0, the rate of change increases, indicating the parabola's widening.
Function: y = x^2 + 4
This quadratic function opens downward (because of the negative coefficient of x^2
), and its rate of change shows deceleration as x
moves away from the vertex at (0, 4).
Equation: y = 50x
If y
represents the total cost of buying x
apples at a constant price of $50 per apple, the rate of change, or slope, is 50. This indicates that the cost (output) increases by $50 for every additional apple (input) purchased.
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