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Each diagram below shows part of the graph of a function - Leaving Cert Mathematics - Question 5 - 2013
Question 5
Each diagram below shows part of the graph of a function. Each of these functions is either quadratic or cubic or trigonometric or exponential (not necessarily in th... show full transcript
Worked Solution & Example Answer:Each diagram below shows part of the graph of a function - Leaving Cert Mathematics - Question 5 - 2013
Step 1
Type of function: Quadratic
Answer
Function:
First derivative:
Second derivative:
Explanation
The graph of a quadratic function is a parabola. Its first derivative is a linear function, represented by graph B, which has a constant slope, demonstrating that the original function changes at a steady rate. The second derivative is a horizontal line (graph I) indicating a constant rate of change of the slope, confirming it's a quadratic function.
Step 2
Type of function: Cubic
Answer
Function:
First derivative:
Second derivative:
Explanation
A cubic function is characterized by its S-shaped curve. Its first derivative reflects this shape, represented by graph D, which has one local extrema. The second derivative (graph II) shows a point of inflection, typical of cubic functions.
Step 3
Type of function: Trigonometric
Answer
Function:
First derivative:
Second derivative:
Explanation
The graph of a trigonometric function oscillates. Its first derivative (graph C) also oscillates but is phase-shifted. The second derivative (graph III) shows the nature of oscillation in a trigonometric function as it indicates the rate of change of the first derivative.
Step 4
Type of function: Exponential
Answer
Function:
First derivative:
Second derivative:
Explanation
An exponential function increases rapidly. Its first derivative (graph A) is also an exponential function but shifted down, while the second derivative (graph IV) indicates the accelerating growth of the original exponential function.