Functions and Graphs

Functions and Graphs

Article Contents

  1. The Coordinate System
  2. Points in the Plane
  3. Plotting Points from a Table
  4. Drawing a Graph
  5. Reading Values from a Graph
  6. Increasing and Decreasing
  7. Intercepts
  8. Examples of Common Graphs
  9. Common Mistakes
  10. Interactive Exercises

1. The Coordinate System {#coordinate-system}

A Cartesian coordinate system consists of two perpendicular number lines:

O x y -3 -2 -1 1 2 3 4 -1 -2 -3 1 2 3 I II III IV
  • x-axis (horizontal): shows the independent variable
  • y-axis (vertical): shows the dependent variable
  • Origin : where the axes cross
  • Quadrants I, II, III, IV: the four regions created by the axes

2. Points in the Plane {#points-in-plane}

Every point is described by an ordered pair :

  • = horizontal position (left/right from origin)
  • = vertical position (up/down from origin)

Examples

O A(2, 2) B(-2, 1) C(-1, -2) D(3, -1)
PointQuadrant
I
II
III
IV

3. Plotting Points from a Table {#plotting-points}

Given a function table, plot each pair as a point.

Example:

x y -3 -2 -1 1 2 3 1 2 4 6 -1 -2 f(x) = x^2 - 2

4. Drawing a Graph {#drawing-graph}

Step-by-step process

  1. Create a table of values (see Functions and Tables)
  2. Set up the coordinate system with appropriate scale
  3. Plot each point
  4. Connect the points:

- With a straight line for linear functions

- With a smooth curve for other functions

  1. Label the graph with the function formula

Tips for choosing -values

  • For linear functions (): 2 points are enough, but use 3 for safety
  • For quadratic functions (): use at least 5--7 points
  • Always include if it is in the domain
  • Choose values symmetrically around the interesting part

5. Reading Values from a Graph {#reading-from-graph}

x y 1 2 3 4 -1 -2 1 2 3 -1 -2 f(2) = 1 x-intercept f(x) = x - 1

To find -- reading the output:

  1. Find on the horizontal axis
  2. Go vertically to the graph
  3. Read the -value at that point

To find when -- reading the input:

  1. Find on the vertical axis
  2. Go horizontally to the graph
  3. Read the -value at that point

6. Increasing and Decreasing {#increasing-decreasing}

A graph shows us where a function is increasing or decreasing:

  • Increasing: the graph goes up as you move right (when grows, grows)
  • Decreasing: the graph goes down as you move right (when grows, shrinks)
x y decreasing increasing

7. Intercepts {#intercepts}

x-intercept (zero of the function)

The point where the graph crosses the x-axis, i.e., where .

y-intercept

The point where the graph crosses the y-axis, i.e., the value .

Example:

  • y-intercept: , so the graph crosses the y-axis at
  • x-intercept: , so the graph crosses the x-axis at

8. Examples of Common Graphs {#common-graphs}

x y y = x y = x^2 y = 2 y = 1/x
Function typeEquationGraph shape
ConstantHorizontal line
LinearStraight line
Direct proportionLine through origin
Inverse proportionHyperbola
QuadraticParabola

9. Common Mistakes {#common-mistakes}

MistakeHow to avoid it
Swapping and coordinatesAlways write -- horizontal first, vertical second
Connecting points with straight lines when the function is curvedUse a smooth curve for non-linear functions
Not enough points for a curved graphPlot at least 5--7 points for quadratics and other curves
Forgetting to label axes and scaleAlways mark the scale and axis names
Plotting instead of The first number is always the horizontal position

10. Interactive Exercises {#interactive-exercises}

Practice graphing skills:

Summary

ConceptDescription
Coordinate systemTwo perpendicular axes (x horizontal, y vertical) with origin O
Point = horizontal position, = vertical position
PlottingCreate a table, plot points, connect them
Reading Go from vertically to the graph, read
x-interceptWhere (graph crosses x-axis)
y-interceptThe value (graph crosses y-axis)