# Graphs of the Antiderivatives of Functions

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Recall that the words **integral** and **antiderivative** refer to the same process

- and they are the reverse of
**derivative**.

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So if we let **F(x)** be the antiderivative of **f(x)**

… … $F(x)= \displaystyle{ \int f(x) \; dx}$

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then it is also true that **f(x)** is the derivative of **F(x)**

… … $f(x) = \dfrac{d}{dx} \Big( F(x) \Big)$

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**BUT** the process of taking the derivative loses a piece of information:

- the value of the constant term is lost when we take the derivative.
- This is why we have to add
**+c**when finding indefinite integrals.

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This means that, when it comes to graphs:

- If we start with the graph of a function,
- we can accurately sketch the graph of its derivative.

**BUT**

- If we start with the graph of a function, $f \big( x \big)$,
- we can accurately sketch the
**shape**of its antiderivative, $F \big( x \big) +c$, - but we can't know
**where**that shape should be placed**vertically**on the Cartesian axes- because we don't know the value of the constant (
**+c**).

- because we don't know the value of the constant (

- we can accurately sketch the

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Therefore the antiderivative, $F \big( x \big) + c$ represents a **family of curves**

- all with the same shape but translated up or down by different amounts.
- There are an infinite number of possible curves $F\big( x \big) + c$,
- due to an infinite number of possible values for
**c**.

- due to an infinite number of possible values for

- ie the
**family of curves**is an infinite set.

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When sketching $F \big( x \big)$, we place the shape of $F \big( x \big)$ somewhere on the axes and call it **an** antiderivative of $f \big( x \big)$.

- I usually try to place it so the antiderivative graph doesn't overlap the graph of the original function (or keep the overlap to a minimum)

- Some people draw it with a y-intercept of $(0, \; 0)$

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- One point (one set of coordinates) that $F \big( x \big)$ passes through is enough to determine the value for
**c**and hence where the antiderivative should be drawn.

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## Features of the Graphs of Antiderivatives

… For $F \big( x \big) = \displaystyle{ \int } f\big( x \big) \; dx$

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### Example 1

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Consider the following graph of a function, $f \big( x \big)$.

- Key features of the graph have been annotated.

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This information can now be used to sketch an **antiderivative** graph, $F \big( x \big)$, (in red).

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