4L - Exploring rates of change with parabolas
Introduction to rates of change:
Rate of change can be defined as how one quantity (\(Q_1\)) changes in relation to another quantity (\(Q_2\)) changing. Mathematically this can be expressed as: \[ROC=\frac{\Delta Q_1}{\Delta Q_2}\] where \(\Delta\) is the "change in". It is important to remember that the rate of change is a quantity.
Rate of change can be defined as how one quantity (\(Q_1\)) changes in relation to another quantity (\(Q_2\)) changing. Mathematically this can be expressed as: \[ROC=\frac{\Delta Q_1}{\Delta Q_2}\] where \(\Delta\) is the "change in". It is important to remember that the rate of change is a quantity.
Qualitative description of rates of change:
Rates of change can be positive, negative or zero:
Rates of change can be positive, negative or zero:
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The average rate of change:
■ The average rate of change is found by calculating the gradient of a line segment joining two points on a graph. \[AvROC=m=\frac{y_2-y_1}{x_2-x_1}\]
4L - Example 1: Calculating the average rate of change with parabolas
Consider the parabola \(y=-x^2-2x+3\) as shown.
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4L - Example 1: Video solution
4L - Example 1: Practice
Question 1: ABC Question 2: ABC 4L - Example 1: Solutions
Question 1: ABC Question 2: ABC |
The instantaneous rate of change:
■ The instantaneous rate of change is found by calculating the gradient, \(m\), of a tangent line to a graph at the desired point. The gradient of the tangent line describes the rate of change at that point or instant.
To approximate the gradient of a tangent to a graph on the Casio ClassPad II, use the following steps:
- Type the equation of the curve into the Main screen.
- Drag the equation into the graph screen to sketch the graph.
- Click: Analysis → Sketch → Tangent.
- Using the hard keypad, enter the \(x\)-value you want that tangent at and hit OK.
- Hitting exe on the hard keypad will bring up the equation (\(y=mx+c\)).
- The value of \(m\) (the coefficient of \(x\)) is the approximate value of the gradient.
4L - Example 2: Calculating the instantaneous rate of change with parabolas (CAS)
Using a CAS calculator, determine the instantaneous rate of change, of \(y\) with respect to \(x\), for the quadratic equation \(y=x^2-3x\) at
(a) \(x=-1\) (b) \(x=1.5\) (c) \(x=2.8\) |
4L - Example 2: Video solution
4L - Example 2: Practice
Question 1: ABC Question 2: ABC 4L - Example 2: Solutions
Question 1: ABC Question 2: ABC |
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Additional Exercises
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Topic Worksheets
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Other Resources
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Determine the average rate of change of \(y=3x^2-6x-2\) between \(x=2\) and \(x=3\)
Determine the average rate of change of \(y=-x^2+3x-6\) between \(x=1\) and \(x=3\)
Teresa went skydiving. The graph below describes Teresa's height (measured in meters) as a function of time (measured in seconds).
Find the average rate of change which the height decreases between 3 seconds and 8 seconds during Teresa's skydive.
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Solutions:
Question 1.1: (a) Average rate of change \(=3\) (b) Average rate of change \(=-1\) Question 1.2: Average rate of change \(=-8\) Question 1.3: Average rate of change \(=5\) Question 2: \[AvROC=\frac{-6-(-4)}{3-1}=-1\] Question 3: Approximately \(55\) \(m\)/\(s\) |
Worksheet A - Calculating rates of change with parabolas
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Worksheet B - Applying rates of change to "ski jumps"
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Khan Academy - Introduction to average rate of change:
Khan Academy: Introduction to average rate of change
Essential question:
What is the average rate of change of a function over an interval? |
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