Lesson 5: Graphs

New Concept

Graphs provide a visual representation of the relationship between two variables: an input and an output. You will learn to plot points from a table of values, graph an equation, and read the graph to analyze the equation's behavior.

What’s next

You'll start by converting bar graphs to line graphs, then master plotting points from equations through interactive examples and a series of practice cards.

Property

Graphs are especially useful for illustrating the relationship between two variables. We could instead place a dot at the top of each bar. If we connect the dots with line segments, we have created a line graph. We often use line graphs to illustrate trends in data over time.

Examples

• A line graph shows a student's test scores over a semester: 85 in September, 88 in October, 92 in November, and 90 in December. The line connecting these points shows a general improvement.

• A line graph tracks the daily temperature for a week. The points might be $15^\circ C$, $17^\circ C$, $16^\circ C$, $19^\circ C$, $20^\circ C$, $18^\circ C$, and $17^\circ C$. The line shows the temperature fluctuations.

Property

To plot points:

• The first variable (the input variable) tells us the location of the point in the horizontal direction.

• The second variable (the output variable) tells us its location in the vertical direction.

Property

Steps for Graphing an Equation.

1. Make a table of values. Choose values for the input variable and use the equation to find the values of the output variable.

2. Choose appropriate scales and label the axes.

Property

Each point on the graph of an equation represents a pair of values for the variables that makes the equation true. To read a graph, find a point on the curve, then trace down to the horizontal axis to find the input value and trace across to the vertical axis to find the corresponding output value.

Examples

• A graph shows a car's journey. To find the distance traveled after 2 hours, find 2 on the horizontal time axis, move up to the graph line, and then move left to the vertical distance axis to read the value, say 120 miles.

• On a graph of the equation $y = x + 5$, to find the value of $y$ when $x=3$, locate 3 on the horizontal axis, move up to the line, and read the corresponding value on the vertical axis, which is 8.