Introduction to the culminating assessment - Math journal

Topic 1 – Polynomial functions

Here are examples of two types of polynomial functions you are familiar with:

Linear: $f\left(x\right)=x+3$
Quadratic: $f\left(x\right)= x^2+4x+3$

Can you see a pattern? Try to think of another type of polynomial function. For example:

Cubic: $f\left(x\right)= x^3+{5x}^3+x-7$
Quartic: $f\left(x\right)= x^4-{3x}^3+x^2-5$

The parts of a polynomial function

A Bold text startpolynomial function Bold text Endhas the form

$f\left(x\right)= a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+...+a_3{x}^3+a_2{x}^2+a_{1}x+a_0$

It may be helpful to search the internet for tutorials explaining the “parts of a polynomial”.

Example:

Let’s go through an example of identifying parts of a polynomial equation. Remember to check your solutions with the suggested answers.

We’ll use the polynomial function $f\left(x\right)= {4x}^5-{7x}^4+{2x}^3-{6x}^2+10x+15$ , to identify the following:

Types of functions:

Rational function:

$y= \frac{f\left(x\right)}{g\left(x\right)}$

Created when one polynomial function, $f\left(x\right)$, is divided by another, $g\left(x\right)$

Note: The polynomial function in the denominator, g(x), must have a degree of at least 1.

Think of an example of a rational function and record it in your notebook. Look at the suggested answers to check your ideas.

Square Root Function:

$y= \sqrt{f\left(x\right)}$

Created by taking the square root of a polynomial function, $f\left(x\right)$

Think of an example of a square root function and record it in your notebook. Look at the suggested answers to check your ideas.

Examples:

Let’s go through some examples of identifying polynomial functions.

Identify the degree and leading coefficient for the polynomial functions.

Now, see if you can match each function included on the left with its correct classification on the right.

Topic 2 – Power functions

Power function:

A power function is the simplest form of a polynomial function:

$f\left(x\right)= {ax}^n$

$a:$ any real number

$x:$ a variable

$n:$ any natural number

Some power functions are given special names based on their degree. Review each of the following power functions, identifying the degree and type of power function. Look at the suggested answer to check your ideas.

1. Identify the degree and the type of power function $y=a$.

2. Identify the degree and the type of power function $y=ax$.

3. Identify the degree and the type of power function $y={ax}^2$.

4. Identify the degree and the type of power function $y={ax}^3.$

5. Identify the degree and the type of power function $y={ax}^4$.

Properties of a function

TheBold text start Bold text EndBold text startend behaviour Bold text Endof a function tells us how the y-values are changing at both horizontal ends of the graph or as the $x$-values get very large or very small:

• $x$-values getting larger in the negative direction
• $x$-values getting larger in the positive direction

Next, you’ll have an opportunity to review each in more detail.

Example: Large Negative and Positive Value of $x$

For each example included in the, “Large Negative and Positive Value(s) of x Practice Exercise” (Opens in new window), identify the end behaviour of each function. Then, check your solutions with the suggested answers that follow.

Bold text startSuggested Answers for Large Positive Value of $x$ Practice ExerciseBold text End
Now, review the suggested answers for large positive value of $x$immediately.

The Bold text startdomainBold text End of a function tells us for which $x-$values the function is possible.

• For example, the function $y=\sqrt{x}$, only positive values of $x$ and 0 are possible, so the domain would be
  • $\left\{xϵ\mathbb{R}|x\ge 0\right\}$, ‘$x$ is an element of real numbers, $R$, such that, |, $x$ is greater than or equal to, $\ge$, 0’

The Bold text startrangeBold text End of a function tells us for which $y$-values the function is possible

• For example, the function $y= x^2-2$, is a parabola that opens upward with vertex (0,-2), so only $y$-values greater than or equal to $-2$ are possible for this function

• $\left\{yϵ\mathbb{R}|y\ge -2\right\}$, ‘$y$ is an element of real numbers, $R$, such that, |, $y$ is greater than or equal to, $\ge$, -2’

Examples:

Let’s go through some examples of identifying the domain and range. Remember to check your solutions with the suggested answers.

Example 1:

Example 2:

Here is another example to try.

Example 3:

Great work! Here is one more example to try.

The $x$Bold text start-intercept(s)Bold text End of a function are the point(s) where the function crosses the $x$-axis or when $y=0$

For example, this function has $x$-intercepts at $(-\mathrm{1,0})$ and $\left(\mathrm{4,0}\right)$

TheBold text start Bold text End$y$Bold text start-interceptBold text End of a function is the point where the function crosses the $y$-axis or when $x=0$

For example, this function has the $y$-intercept at $\left(\mathrm{0,2}\right)$

Examples:

Let’s go through another example of identifying the $x$- and $y$-intercept(s). Select each tab to learn more.

Now, you give it a try. The answers are provided so that you can check your work.

Domain, Range, and End Behaviours Practice Exercise

Using Geogebra or another free online graphing tool, investigate how the domain and end behaviour changes for positive and negative leading coefficients. Then use the, “Domain, Range, and End Behaviours Suggested Answers” (Opens in new window) to check your ideas.

Power functions

Review the success criteria

Rate your understanding on a scale of 1 to 5, where 1 = “I am confused” and 5 =”I have a thorough understanding.

If you rated your understanding below 3, take some time now to think about how you will work to improve your rating where it is needed.

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