AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsFunctions & Rates of ChangeGraphs & Zeros of Functions

Graphs & Zeros of Functions (College Board AP® Precalculus): Revision Note

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Graph basics

What is the graph of a function?

  • The graph of a function is a way of displaying a set of input and output pairs for the function

    • E.g. for every point open parentheses x comma space f open parentheses x close parentheses close parentheses on the graph of y equals f open parentheses x close parentheses

      • the y coordinate f open parentheses x close parentheses shows the output value of the function when the input value is x

Graph with x and y axes showing a polynomial function. Point marked at (19, 28) with label: "The point (19, 28) is on the graph of f, therefore f(19) = 28".
The connection between points on a graph and input-output pairs

  • A graph shows how the function's input and output values vary

    • The pictorial nature of a graph can make this much easier to see in a graph than in, say, a table of input and output values

  • A graph does not need to be based on an explicit equation representing the function

    • A verbal description of the way phenomena change together can be used as the basis for constructing a graph

    • See the Worked Example

What is the rate of change of a linear graph?

  • A linear graph has the form of a straight line

  • The rate of change of the graph is equivalent to the slope of the straight line

    • A positive slope indicates an increasing graph

      • As x increases, f open parentheses x close parentheses also increases

    • A negative slope indicates a decreasing graph

      • As x increases, f open parentheses x close parentheses decreases

    • A steeper slope shows a faster rate of change

    • A horizontal line (slope = 0) indicates no change

      • As x increases, f open parentheses x close parentheses stays the same

Line graph showing five line segments, labeled according to slope of segment: increasing faster, increasing slower, no change, decreasing faster, decreasing slower.
Rates of change of linear graphs

Zeros of a function

What is a zero of a function?

  • The graph of a function intersects the x-axis when the output value is zero

    • bold italic y bold equals bold 0 at those points

    • The input values that correspond to those points are called the zeros of the function

Graph of a polynomial curve intersecting the x-axis at -3, 1, and 4, with peaks and troughs, on a Cartesian grid with x and y axes.
Example of a function with zeros at x=-3, x=1 and x=4

How can I find the zeros of a function?

  • You may be able to read the zeros off of an accurate graph of the function

    • Look for the x coordinates of the points where the graph crosses or touches the horizontal axis

  • If you know the equation of a function, then you can find the zeros

    • by setting the equation equal to zero

      • and solving for x

    • E.g. f open parentheses x close parentheses equals x squared minus 2 x minus 8

      • x squared minus 2 x minus 8 equals 0

      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses equals 0

      • x equals negative 2 comma space space x equals 4

    • The zeros of f are x equals negative 2 and x equals 4

Worked Example

A pump is used to fill a water tank, which is initially empty. When the pump is running, the rate at which the volume of water in the tank increases is constant. During the first three hours the pump is running on its highest setting. After a noise complaint from a neighbour, the pump is temporarily switched off. The pump is eventually switched back on, using a lower setting that pumps the water more slowly. The pump is run on this setting until the tank is full. The entire process of filling the tank takes eight hours. Which of the following graphs could depict this situation, where time, in hours, is the independent variable, and the volume of water in the tank, in gallons, is the dependent variable.

Four graphs labelled A, B, C, D, showing volume in gallons over time in hours, each with different line segments indicating changes in volume.

Answer:

Consider the information in the question

  • "The entire process of filling the tank takes eight hours" rules out graph (A)

    • because the process in that graph takes 10 hours

  • "The pump is temporarily switched off" and "The pump is eventually switched back on" mean that there should be a horizontal section of the graph where the volume isn't changing

    • The question doesn't say how long the pump was off, though, so that can't help you to choose among the remaining graphs

  • "During the first three hours the pump is running on its highest setting"

    • This means the initial segment of the graph should go from 0 to 3 on the horizontal axis

    • This rules out graph (B), because the initial segment there lasts for 4 hours

  • "During the first three hours the pump is running on its highest setting" and "The pump is eventually switched back on, using a lower setting that pumps the water more slowly" together give you the final necessary clue

    • The pump is pumping water more quickly at the start, and more slowly at the end

    • This means the graph should go up more steeply at the start, and less steeply at the end

    • This rules out graph (D), and would also rule out graphs (A) and (B)

After considering all the information, only one graph is left which matches all the conditions in the description

Graph (C)

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.