Lesson 5 - Characteristics of Functions
Unit 1
Chapter 5 - Characteristics of Functions
LESSON 1: UNIT INTRODUCTION
- Distinguish a function from a relation.
- Represent linear and quadratic functions using function notation.
- Describe the domain and range of a function using the proper notation.
- Determine the numeric or graphical representation of the inverse of a linear or quadratic function.
- Identify the roles of the parameters a, k, d, and c in functions of the form:
y = af(k(x – d))+c.
Solving problems involving Quadratic Functions
- Determine the number of zeros of a quadratic function.
- Determine the maximum or minimum value of a quadratic function.
- Solve problems involving quadratic functions arising from real-world applications.
- Investigate the families of quadratic function, their transformational relationships and algebraic representations.
- Solve systems of linear and quadratic functions.
Determining Equivalent Algebraic Expressions
- Simplify polynomial expressions by adding, subtracting, and multiplying.
- Simplify radicals.
- Simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restriction(s).
- Determine if two algebraic expressions are equivalent.
- There is a certain beauty about systems and order. They help us to better understand concepts and events so that we can approach them with predictably. The systems we are studying are called functions. To fully understand them, we will explore their properties and characteristics, and organize and express them in meaningful ways.
- Whether we approach a function by looking at it algebraically, graphically, or by considering its properties, we should handle and apply functions with proficiency and purpose.
- You are already familiar with graphing functions and some formatting standards. Proceed attentively through this unit to refine your communication skills in the language of mathematics.
- Make sure everything is clearly defined and that your solutions are developed fluidly and purposefully. The following lessons will introduce you to the concepts necessary to understanding characteristics of functions.
Lesson 2.1: Review and Solving Quadratic Functions
- Quadratic functions are not new to us. We learned about quadratics in grade 10. There are three different forms of a quadratic: the general form, the vertex form (also called standard form), and the factored form.
- When we graph a quadratic function, we get a U-shaped graph called a parabola. We often see this parabola shape in real life. For example, when a baseball is hit, it flies through the air following a parabolic path.
- Quadratic functions are also used in many other fields, including architecture and design, business, and finance.
- In this form, the a value tells us the direction of opening of the parabola. If the a value is positive, the parabola opens up. If the a value is negative, the parabola opens down.
- The a value also tells us whether the function is stretched or compressed. If a is greater than 1, the graph is stretched vertically by a factor of a. If 0 < a < 1, the function is compressed vertically by a factor of a.
- In this form, the a value tells us the direction of opening of the parabola. If the a value is positive, the parabola opens up. If the a value is negative, the parabola opens down.
- The a value also tells us whether the function is stretched or compressed. If a is greater than 1, the graph is stretched vertically by a factor of a. If 0 < a < 1, the function is compressed vertically by a factor of a.
- This form of the quadratic also tells us the location of the vertex of the parabola. The vertex is located at (h, k)
- In this form, the a value tells us the direction of opening of the parabola. If the a value is positive, the parabola opens up. If the a value is negative, the parabola opens down.
- The a value also tells us whether the function is stretched or compressed. If a is greater than 1, the graph is stretched vertically by a factor of a. If 0 < a < 1, the function is compressed vertically by a factor of a.
- This form of the quadratic function also tells us the x-intercepts (also called roots or zeros) of the parabola. The x-intercepts are located at (r, 0) and (s, 0).
- We can manipulate between various forms of a quadratic to get the form we need. The following flow chart shows the methods used to convert between the three forms.
- There are no arrows between the factored form and the vertex form. This is because we do not have a direct method between these two forms. Instead, we first need to apply the distributive law to convert to standard form, then we can apply the appropriate method to convert to the form we need.
Converting Between Forms - Distributive Law
As we see in the flow chart above, we can convert to the general form of a quadratic by applying the distributive law. If you don’t recall the distributive law, you can review it by watching the following video.
Converting Between Forms - Standard Form
We can convert to the vertex form by using the completing the square method. The following examples will help you remember how to convert from the vertex form to the general form.
Step by Step
Solving Quadratic Functions
- Solving quadratic functions means finding the x-intercepts (also called roots or zeros). These are the values of x where the graph of the quadratic function touches the x-axis. Since the x-axis is located where y= 0, we are solving the quadratic function for y= 0.
- If we graph a quadratic function, we will produce a parabola. There may be two x-intercepts, one x-intercept, or no x-intercepts at all depending on where the parabola is located on the Cartesian plane.
- If there are two x-intercepts, it means there are two distinct real solutions.
- If there is one x-intercept, it means there is a double solution.
- If there are no x-intercepts, it means there are no real solutions.
Based on the graph, do each of these functions have two distinct x-intercepts, a double x-intercept, or no real x-intercepts?
The graph of a quadratic function doesn’t just tell us how many x-intercepts there are. We can also determine the x-intercepts by inspecting the graph. Inspecting means that we will look at what points along the x-axis the graph crosses, as shown in the example below.
Example: What are the x-intercepts of the quadratic shown in the graph below?
Common Factor:
A common factor is a number (e.g.2), a variable (e.g.x), or an expression (e.g.t-1) that evenly divides all of the terms in an expression.
Common Factor the following polynomial
Factoring by grouping can be used to factor quadratic functions in their standard form by grouping items together and finding common factors.
- In this form, the a value tells us the direction of opening of the parabola. If the a value is positive, the parabola opens up. If the a value is negative, the parabola opens down.
- The a value also tells us whether the function is stretched or compressed. If a is greater than 1, the graph is stretched vertically by a factor of a. If 0 < a < 1, the function is compressed vertically by a factor of a.
Factoring Methods – Differences of squares
Differences of squares are expressions of the form a2 – b2
Differences of squares can easily be factored like this:
Factoring Methods – Simple trinomials
Simple trinomials are functions of the form
The following example demonstrates how to determine the factored form of a simple trinomial.
Factoring Methods - Complex trinomials
Complex trinomials are functions of the form
Follow these steps to factor a complex trinomial.
Once the quadratic has been rewritten into the factored form, we can solve for the x-intercepts by applying the zero product property.
As you may also recall, another method for solving quadratic functions is the quadratic formula. The quadratic formula allows us to solve quadratic functions without factoring and without having to set each of our factors equal to zero and solve.
A part of the quadratic formula called the discriminant can also be used to determine the number of solutions for a quadratic function.
LESSON 2.3: FAMILIES OF FUNCTIONS AND MODELLING WITH QUADRATIC FUNCTIONS
- In some ways, quadratic functions are similar to human beings. Humans can be grouped into larger identifying categories, such as the country in which they live. Within a given country are subgroups called provinces, states, counties, and so on. Within these subgroups are further identifying subgroups called families.
- Human families share certain traits. Here, many will think of DNA, but families also share other attributes, such as a common heritage, which is often referred to as a person’s “roots.”
- Quadratics are part of a larger group of functions called polynomials. Quadratic functions also have families. The members of a family of quadratic functions all have the same x-intercepts. You may recall that x-intercepts are sometimes called solutions or roots—what a coincidence!
- Each of the quadratic functions in the graph above pass through the same two x-intercepts. All of these functions are part of the same family of functions.
- All of these functions, written in the factored form, have the same x-intercepts. The only difference between these functions is the a-value. This gives us a definition for families of quadratic functions.
Families of Quadratic Functions
Modelling with Quadratic Functions
- Quadratic functions can be used to model real-world scenarios. Sometimes we are given the function that models a situation, but that isn’t always the case. In some cases, we may only know details about the situation and have to use them to determine a function that models the situation.
- What information do we need to know in order to determine the function that models our situation?
Modelling with Quadratic Functions
Before we start trying to model a real-world scenario, let’s start by looking at what we know about quadratic functions
- First, we know that quadratic functions can be written in three forms:
- We also know that the graph of a quadratic function has a parabola shape and that we can identify points on the quadratic function by inspecting the graph.
Modelling with Quadratic Functions
Before we start trying to model a real-world scenario, let’s start by looking at what we know about quadratic functions
- First, we know that quadratic functions can be written in three forms:
- We also know that the graph of a quadratic function has a parabola shape and that we can identify points on the quadratic function by inspecting the graph.
- In Figure 1, we can identify the x-intercepts. We can substitute the x-coordinates of the x-intercepts into the factored form, for r and s, but we still need to determine the value of a. By substituting the third point into the function, for x and y, we can solve for a.
Let's solve for the quadratic function that models the graph in Figure 1.
The Vertex-Point Method
- Not every quadratic function has x-intercepts. How can we solve for the function in these cases?
- In Figure 2, we can identify the vertex from the graph. We can substitute the vertex into the vertex form for h and k, but we still need to determine the value of a. By substituting another point into the function, for x and y, we can solve for a.
Let's solve for the quadratic function that models the graph in Figure 2.
Discussion Topic
LESSON 2.4: SOLVING LINEAR AND QUADRATIC FUNCTIONS
Solving Linear and Quadratic Functions
- In many applications of functions, we often need to determine when two functions intersect. Solving for the point or points of intersection of two or more functions is called solving a system of functions.
- A linear function and a quadratic function can intersect three different ways.
- Two Points of Intersection
- One Point of Intersection
- No Points of Intersection
- A linear function and a quadratic function can intersect at two points, as this graph shows:
- A linear function and a quadratic function can intersect at a single point, as this graph shows:
- A linear function and a quadratic function may not intersect at all, as this graph shows
- Solving by inspection is a simple way to solve for the points of intersection of two functions. To solve by inspection, we need to first graph our functions on the same set of axes, like this:
- While solving by inspection may be very quick and easy in some cases, it’s not always the easiest method.
In the graph above, the points of intersection are harder to identify. These points of intersection are obviously going to contain decimal values, so we will need to use a more accurate method to solve for the points of intersection.
- To solve a system of equations algebraically, we need to know the functions.
- As we just saw in the previous graphs, the points of intersection are located at points where the graphs of the two functions are equal. In other words, if we input the same x-value into both functions, our outputs will also be the same.
- The following example will demonstrate how we can solve functions algebraically.
Lesson 5: Rational Expressions
What is a rational expression?
- A rational expression is an expression that contains a fraction, where the numerator and denominator are both polynomials.
- We can manipulate rational expressions in the same way as fractions. However, since a rational expression involves variables, we need to consider the restrictions on the variables.
- As you know, the denominator of a fraction can contain any number except zero, because a fraction such as 21/0 is not defined. The same thing applies to rational expressions: the denominator cannot be zero. When asked to simplify a rational expression, you need to find the restrictions first.
- To find the restriction, ask yourself what values of the variables make the denominator equal zero?
Simplifying Rational Expressions
- Simplifying a rational expression is similar to simplifying a fraction. When simplifying a fraction, we look for common factors in the numerator and denominator that will divide out. To simplify a rational expression, we also need to determine common factors in the numerator and denominator that will divide out.
- In order to determine the common factors, the polynomials in the numerator and denominator need to be in factored form. If they are not in factored form, we will need to factor them by using our factoring techniques.
Multiplying Rational Expressions
- When multiplying rational expressions, we follow the rules for multiplying fractions. That is, the numerator is multiplied by the numerator and the denominator is multiplied by the denominator.
- After multiplying, remember to always simplify the rational expression.
Adding and Subtracting Rational Expressions
- As mentioned previously, rational functions are fractions. They therefore behave like fractions, meaning that when we add or subtract two rational expressions, they must have a common denominator.
- If two rational expressions do not have a common denominator, we will need to multiply the numerators and denominators by factors so that they have a common denominator.
The following example demonstrates how to add two rational expressions.
