Section 3 – Using Graphs with Linear Functions

Objectives

Use linear graphs to solve equations
Draw graphs of linear functions
Write linear functions from graphs

Linear graphs are a useful tool for solving equations, providing visual representations, or making predictions about data. There a multiple ways to graph linear equations, we will explore the following two techniques:

(1) Generate a table of points (ordered pairs), plot them, and connect.

(2) Use the equation to identify the slope and y – intercept, then graph.

The skills and techniques used for evaluating and graphing linear functions can be extended to non-linear functions with some modifications. Linear functions are the simplest example of algebraic functions, and will develop the skills needed to move on to more advanced mathematics.

Solving Linear Functions Graphically

Linear functions can be solved using their graphs to either solve for the output of a function for a given input value, or to find which input values will produce a predetermined output. The difference simply comes down to which coordinate axis should we start.

Ex 1: Evaluate for a given input

The graph of a function is shown on the left. Use the graph to evaluate . This means that you must locate x = 2 on the horizontal axis, then move vertically up until you touch the graph and approximate the location of the y – coordinate. The image below shows the result.

Ex 2: Solve for given a predetermined output

Using the graph of again, but this time determine for which x – coordinate does . The first step is to restate the question. Since is the same as , the question can be reworded as “What is the x-coordinate when the y-coordinate is -2?”. This means that you must locate y = – 2 on the vertical axis, then move horizontally to the right until you touch the graph and approximate the location of the x – coordinate. The image below shows the result.

Drawing the Graph of a Linear Function

Graphing functions is an important part of mathematical analysis. As we learn to draw graphs for different types of functions, linear and non-linear, we are able to anticipate data trends just by becoming familiar with the shape of a graph and the direction in which it will advance. As previously mentioned, we will focus on two methods for drawing the graph of a linear function.

(1) Plot a Table of Values (2) Slope-Intercept Graphing

Ex 1: Graph a Linear Function Using a Table of Values

The technique of plotting points from a table of values can be used to graph any function whether linear or non–linear. This is beneficial because if you do not know the shape or direction of a graph you can ascertain such information simply by plotting more points.

Consider the linear function . We know that it is a linear function because earlier we studied that linear functions have equations where the variable exponent is positive one and where variable do not multiply one another, and where variables are not in the denominator of a fraction or under a radical.

Since is a linear function we know the graph is a straight path, therefore we really only need two points to graph a line. However, it is suggested that we graph more than two points because this will often catch a miscalculation on our part and it will extend the precision of our graph through the coordinate plane.

Create a Table for f (x) = 2x – 1

Step 1: It is common to use the x–values = {–2 , –1 , 0 , 1 , 2}

Step 2: Use the function to calculate the corresponding y–values.

Step 3: Interpret the table of values as points to graph in the coordinate plane.

(–2 , –5) (–1 , –3) (0 , –1) (1 , 1) (2 , 3)

Step 4: Connect all the points with a line. If any of the points are off the line then you should double check your calculations for that point.

Slope–Intercept Graphing

When a linear function is expressed in the form , it is referred to as “slope–intercept form”. This is because the m–value is the slope of the linear equation and the b–value is the y–intercept. Therefore, the equation has a slope = and a y–intercept = –4. Due to the fact that this information is easy to extract from the equation, we can rapidly graph the linear function without generating a table of values.