MATCHING GRAPHS OF FUNCTIONS AND THEIR DERIVATIVES WORKSHEET

Six graphs of functions are below, along with six graphs of derivatives. Match the graph of each function with the graph of its derivative

Questions :

Derivative graph

Problem 1 :

Solution :

The graph is a raising line, it will represent the function in the form ax + by + c = 0

While finding the derivative of the function, dy/dx = constant

Graphical form of the derivative must be a horizontal line.

Problem 2 :

Solution :

The graph of original function is parabola. It must be in the form of y = x²

Its derivative will be dy/dx = 2x

By analyzing the graph of original function,

• In (-∞, 0), the original function is increasing, then in the derivative graph should be above the x-axis in this interval.
• (0, ∞), the original function is decreasing, then in the derivative graph should be below the x-axis in this interval.

Problem 3 :

Solution :

The graph of original function is absolute value function. It must be in the form of y = a|x - h| + k

Derivative of absolute value function,

y' = a(x - h) / |x - h| 

On x = h, there is a hole.

• In (-∞, 0), the original function is decreasing function
• In (0, ∞), the original function is increasing.
• At x = 0, we have sharp edge, so derivative is not defined at x = 0

Problem 4 :

Solution :

Lim x --> 0^- f(x) ≠ Lim x --> 0⁺

Then the derivative is not defined at x = 0. Since the original function is raising and falling line, its derivative graph must be horizontal line.

Problem 5 :

Solution :

The graph of the original function is looks like a graph of cubic function. Then its derivative graph must be a graph of quadratic function. By observing the graph more clearly, it is increasing function. So, the entire derivative graph must be above the x-axis.

Problem 6 :

Solution :

By observing the graph, there are two pieces and at x = 0 we have sharp edge.

At x = 0, the derivative is not defined and the original graph is increasing entirely. Then its derivative graph must be above the x-axis.