A Parabola Without x Intercepts…Inconceivable!

Graph of a Parabola without x intercepts

In a previous Math-FAQ, we looked at the different parts of a parabola. Based on this information, you know that to find the x intercepts of a parabola we need to solve a quadratic equation. When we solve a quadratic  equation to find the x intercepts of the graph, you might expect to always have solutions. But as the Math-FAQ below shows, this is not always the case.

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How Can a Parabola Have No x Intercepts?

Students are often surprised when they graph a parabola a notice that the parabola has no x intercepts.

But as the graph above shows, parabolas do exist that do not cross the x axis.

However, suppose you do not have the graph of  y = x2 +2x+3 available. How could you use the equation to determine whether this parabola has any x intercepts?

Let’s start by following the usual process for finding x intercepts of any graph. Set y = 0 to get the equation

This is a quadratic equation with a = 1, b = 2, and c = 3. To solve this equation, we need to use the quadratic formula:

Now put in the values for a, b, and c. This gives us

This might set off alarms in your mathematical brain!

How can you take the square root of a negative number? For numbers graphed on a real number graph, you can’t. That is why our graph above has no x intercepts. However, if we expand our knowledge of numbers to complex numbers, we can write out a solution to the quadratic equation .

In complex numbers, is defined to be equal to the letter i. To evaluate the square root above, think of it as

Since i = and , we can simplify our square root as

And our solution to the quadratic equation as

In short, the quadratic equation has a solution that uses i. Since our graph does not allow for this type of number, it shows no x intercepts.

Section 3.2 Dimensional Analysis

Since many measurements in the United States are in the English system, you may often be required to convert from English units like pounds to SI units like kilograms. Luckily, we can use facts (like 1 kilogram is approximately 2.2 pounds) to convert between different measurement systems.

Distance sign with 10 miles 16 kilometers
Road sign with kilometers and miles

Our objectives for this section are to

  • use dimensional analysis to convert from one system of measurement to another system of measurement.

Use the resources below to help you accomplish these objectives.

Section 3.2 Workbook (PDF) – 9/7/19

Section 3.2 Practice Solutions

Chapter 3 Practice Solutions (PDF)

College Mathematics Resources

Full Chapters of Workbooks 

Chapter 1 Statistics (PDF) (Word) – 9/16/19

Chapter 2 Probability (PDF) (Word) – 9/23/19

Chapter 3 Units of Measurement (PDF) ((Word) – 9/7/19

Chapter 4 Modeling (PDF) (Word) – 11/2/19

Chapter 5 Consumer Math (PDF) (Word) – 11/2/19

 

Worksheets

Solving Equations (PDF) (Word) – 9/9/19

More Section 1.4, 1.5, 1.6 Practice (PDF) (Word) – 9/9/19

Probability Practice (PDF) (Word) – 9/25/19

 

  • 11/3/19 – Updated solutions to 4.3.3, 4.3.4, 4.4.5 and 4.4.6,
  • 11/2/19 – Added solutions to Chapter 5
  • 10/31/19 – Updated Section 4.4 to include solving exponential equation by converting to log form. Also added new calculator pics from TI-84 Plus SE.
  • 10/4/19 – Corrected answer on Practice Problem 4 in Section 2.4 Solutions
  • 9/23/19 – Corrected typo in Guided Example 9 in Section 2.1. Corrected solution to Practice Problem 7 in Section 2.1. Corrected solution to Practice Problem 6 in Section 2.2.
  • 9/16/19 – Corrected solution to Practice Problem 7 in Section 1.6
  • 9/7/19 – Uploaded most recent files

 

 

Section 1.6 The Normal Distribution

Picture of bell curve
Original creator David Remahl

You have probably heard of the “bell” curve, but you may not be familiar with what is really is. Some students may ask their instructors, “Do you grade on a curve?” Before you ask that question, make sure you understand what it is and whether it may be to your advantage.

The normal distribution (or bell curve) refers to a histogram that looks like the graphic above. It indicates how data is distributed with respect to each other. As you might guess, the peak of the curve corresponds to the mean and its spread matches the standard deviation.

In this section, you learn about the normal distribution and how to apply it to everyday problems. Our objectives are to

  • describe the basic properties of the normal distribution,
  • apply the 68-95-99.7 rule to a set of data,
  • relate the area under a normal curve to z-scores,
  • convert between raw scores and z-scores, use z-scores to compare data.

Armed with this information, you can utilize the normal distribution to answer questions about the relationship of a specific data value to the data values as a whole.

Use the workbook and videos below to help you master these objectives. Make sure you go through the practice problems since they are reflective of the types of problems you might find on a quiz or exam.

Section 1.6 Workbook on the Normal Distribution (PDF) (7-31-19)

Section 1.6 Practice Solutions

Chapter 1 Practice Solutions (PDF)(9-17-19)

Video