This lesson video Not available at this time available video coming soon

Let's say you have a function. What does it look like? How do you graph it? In this lesson, we'll learn how to graph functions by determining and plotting points.

Imagine it's a dark and stormy night. You're lost in the woods. You sense there's something chasing you. You start to run, but you trip over a fallen branch. You turn over and look behind you. And there, in a flash of lightning, is this: f(x).

Wait, hold on. You don't have to be afraid of functions. I know - they look scary. Where'd that f come from, and what does it mean? And, why is it holding that x prisoner? Functions have intimidated math students for as long as there have been bad stories that start with it being a dark and stormy night.

But, in this lesson, we're going to see that functions are just like any other equation. And, in fact, it's very simple to make a graph of a function.

First, let's get a few things straight. A function is a rule that uses an input to produce an output. Let's say you see f(x) = 2x. That just means that for each value of our input, x, our function, f, produces an output of 2x.

So, if we input 5, our output is 5*2, or 10. If we input 7, our output is 7*2, or 14.

So, f(x) = 2x is actually similar to y = 2x. Again, if x = 5, y = 10. In this equation, y is a function of x. The value of x determines the value of y.

Now we can get into graphs. The graph of a function is a visual representation of all of the points on the plane of (x, f(x)). Note that f(x) is taking the place of y in that ordered pair. This graph we're making is just a collection of points.

You know that famous painting by Seurat? The one that looks like a normal picture, but when you look closer, you see that it's really a collection of countless tiny dots? A line on the graph is the same thing. It's just an infinite number of dots.

To graph a function, we take that idea and follow a simple three-step plan. Let's try this out with this function: f(x) = 3x - 4.

Okay, let's start. Step one: Find ordered pairs. Your ordered pairs are your dots. If you could find an infinite number of them, you'd have your graph. But, we don't have time for that here, though that could be a fun project.

We're just going to fill in the chart below.

A chart is helpful for finding ordered pairs.

In the x column, let's pick some numbers. It doesn't matter too much what we pick, but we want to make sure that if this line curves, we check enough numbers to notice that. Let's use -1, 0, 1, 2.

To find the ordered pairs, we plug our x values into the function. Start with -1. Just plug that into our function: f(-1) = 3(-1) - 4. So, our f(x) value here is -7. Let's add that to our chart. That means our first ordered pair is (-1, -7).

0 is next. f(0) = 3(0) - 4. That's just -4.

Next is 1. f(1) = 3(1) - 4. That's -1.

Finally, there's 2. f(2) = 3(2) - 4. That'll be 2.

Okay, we have our points. Now comes Step two: Plot the points. Let's move over to the coordinate plane and plot these ordered pairs.

First is (-1, -7), and then there's (0, -4). If you know you have a straight line, you really only need two points. But, we're being cautious and plotting four points.

So, there's (1, -1), and there's (2, 2).

Step two is to plot the ordered pairs.

That leaves our final step: Step three: Connect the dots.

You probably practiced this in preschool. Did you know then that connecting the dots would be a practical skill in advanced math?

Anyway, before we connect the dots, let's make sure we know what they're doing.

Some functions have curves. You'd be able to tell because a straight line wouldn't go through all of our points. But, if we draw a line through our points below, it does. So, we have our line. And, that's the graph of f(x) = 3x - 4.

Now that we know the three steps, let's get some practice.

First, let's try f(x) = -3x + 6. Remember step one? Find ordered pairs.

Let's make a chart. Okay, now pick some values for x. How about we try even numbers? -2, 0, 2 and 4.

Chart for practice problem 1

Next, plug them into our function. f'(-2) = -3(-2) + 6. That'll be 12.

f(0) = -3(0) + 6. That's just 6.

f(2) = -3(2) + 6. That's going to be 0.

Finally, f(4) = -3(4) + 6. That's -6.

I'm noticing a pattern here.

Okay, step two: plot the points. There's (-2, 12) below. There's (0, 6)

Points for practice problem 1 Finally, step three: connect the dots. This one is definitely a straight line. And, here it is below!? Line for practice problem 1 This is the graph of f(x) = -3x + 6. Let's do a trickier one: f(x) = x^2 + 2. Okay, step one: find ordered pairs. Since x is squared, let's make sure we use both positive and negative numbers. Let's use -4, -2, 2 and 4. So, what's f(-4)? f(-4) = (-4)^2 + 2. That's 18. Next is -2. f(-2) = (-2)^2 + 2. That's 6. Then, 2. f(2) = 2^2 + 2. That's also 6. And, since we're squaring x, we know that 4 will be the same as -4: 18. Now step two: plot the points. Okay, there's (-4, 18) below. There's (-2, 6). There's (2, 6), and there's (4, 18). Points for practice problem 2 Finally, we connect the dots. Now, you can probably tell that this isn't going to be a straight line. This line curves. We can see it curves up on the left below and also up on the right below, but how do we know where it starts? That will be the smallest possible value of y. The lines for practice problem two will be curved. Since we're squaring x, the smallest y value will be when x is 0. Why? As we saw, when we square a negative number, we get a positive number. Therefore, every negative x value will give us a larger y value than 0. So, let's check what y is when x= 0. f(0) = 0^2 + 2. It's 2! So, that's the bottom of our graph, (0, 2). If we connect the dots, we get a parabola, seen below! To finish graphing this function, we need to find the smallest possible value of y. This example is a good reminder that you can always try more points if you're not sure what the graph will be. Just like that painting, sometimes you need a lot of dots to be sure. Hopefully, you don't need as many as Seurat, though. ## Lesson SummaryIn summary, when we want to graph a function, we want to make a visual representation of all of the points on the plane of (x, f(x)). We follow three simple steps. Step one: find ordered pairs. Find as many as you want, though I recommend at least four or five. Step two: plot the points. See what they look like on the graph. Then step three: connect the dots. Channel your inner preschooler and graph your function! |

Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative

Linear Equations: Intercepts, Standard Form and Graphing

How to Find and Apply The Slope of a Line

Graphing Undefined Slope, Zero Slope and More

Equation of a Line Using Point-Slope Formula

How to Use The Distance Formula

The formula for the midpoint of a line

By rearranging a quadratic equation,

How to Graph Cubics, Quartics, Quintics and Beyond

- What are the Different Types of Numbers?
- How to Build and Reduce Fractions
- What is a Decimal Place Value
- Statistical Analysis with Categorical Data
- What is a Variable in Algebra?
- What Are the Five Main Exponent Properties?
- What is a Linear Equation?
- What is an Absolute Value?
- What are Polynomials, Binomials, and Quadratics?
- How to Add and Subtract Rational Expressions
- Perimeter of Triangles and Rectangles
- Properties of Shapes: Rectangles, Squares and Rhombuses
- What Is a Number Line?
- Graph Functions by Plotting Points
- What is a Linear Equation
- How to Write Well: What Makes Writing Good?
- What Are Literary Motifs?
- How to Evaluate Reasoning
- Reading Comprehension Techniques for the GMAT
- How to Organize an Essay
- How to Structure Sentences in an Essay
- Using Rhetorical Skills to Write Better Essays
- How to Write a Great Essay Quickly
- GMAT Registration Information

- SAT: Identifying Errors in Sentence Structure
- 5 Easy Ways to Prepare for the GAT
- Problem Solving
- How to Proofread an Essay for Spelling and Grammar
- Experts Discuss How to Prepare for NTS GAT
- What is a Variable in Algebra?
- Series
- Structure of the SAT Math Section Structure, Patterns and Scoring
- Problems on Trains
- How to Solve a Rational Equation
- What is a Function: Basics and Key Terms
- Rational Exponents
- Choose The Right Employer
- Picture Pattern Mirror and Water Images
- How to Crack NTS GAT - B?

EntryTest.com is a free service for students seeking successful career.

CAT - College of Admission Tests. All rights reserved. College of Admission Tests Online Test Preparation The CAT Online