MTH 1320: Pre-Calc Resource Week of 5/2/2022

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MTH 1320: Pre-Calc Resource

Week of 5/2/2022

By Jordan Cook

Howdy. This resource focuses on material covered in the last week of classes. This is going to be a brief review of everything we have learned so far. This is not a complete review, so be sure to check the previous resources on topics you aren’t as solid on. Don’t forget to come to Group Tutoring on Mondays at 5:15 in room 75 of the basement of Sid Rich! Our last session will be May 2, 2022.

Keywords: Final Review

Topic of the Week:

Final Review!

Highlight #1: Functions

A function is a mathematical assignment of an x-value to only one y-value. You can have different x’s with the same y, but never a y with different x’s. You can test this through the vertical line test, meaning that there is only one point at each vertical section.

A common type of function is a linear function. The graph of a linear function is always a straight line and therefore can be written in slope-intercept form:

𝑓(𝑥) = 𝑚𝑥 + 𝑏

where 𝑚 is its slope or rate of change and 𝑏 is its y-intercept, the value at which it crosses the y-axis. The slope of a linear function determines whether it is increasing, decreasing, or constant. For increasing functions, the outputs increase with the inputs, and the graph has a positive slope. For decreasing functions, the outputs decrease as the inputs increase, and the graph has a negative slope. A constant function (𝑓(𝑥) = 𝑐) outputs only one value, no matter the input. Its graph is therefore a horizontal line with a slope of zero.

Another function you will commonly see are quadratic functions. A quadratic function is “a function of degree two”. Its graph is a parabola. For example, the toolkit function 𝑓(𝑥) = 𝑥2 is the basic quadratic function. The general form of a quadratic function is

𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐,

where 𝑎, 𝑏, and 𝑐 are real constants and 𝑎 ≠ 0. (If 𝑎 were zero, the function would not be of degree 2, and would be a simple linear function!) You can find where the equation intercepts the x-axis by using the quadratic formula:

𝑥 =−𝑏 ± √(𝑏! − 4𝑎𝑐)

2𝑎

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Here is an example of what the graph would look like of a quadratic equation:

It is often easier to visualize a parabola when its equation is in standard form:

𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘,

where (ℎ, 𝑘) is the vertex, or lowest/highest point, of the parabola.

The next type of function we went over was the polynomial function. Polynomials are very common in all math! They are smooth, have no points or corners (also known as cusps) and continuous, with no breaks. For example, the absolute value function, 𝑓(𝑥) = |𝑥|, is continuous, but it is not a polynomial because its graph comes to a sharp point at 𝑥 = 0. It is also important to remember that a polynomial’s domain is all real numbers. To find the x-intercepts or zeros of a polynomial, you will often need to factor it. Factoring higher order polynomials is similar to factoring some quadratic functions. Some important factoring methods are:

• Pulling out the greatest common factor (largest number or power of 𝑥 that is in all the terms) • Factoring binomials that are differences of squares (e.g. 𝑥2 − 9 = (𝑥 + 3)(𝑥 − 3)) • Grouping the terms into groups of two, pulling out the greatest common factor of each group, and pulling out the

common factor between the groups (for four-term polynomials)

We might also need to divide polynomial functions with other polynomial functions. Please refer to the week 6 resource, as that goes into detail about this extensive process.

Asymptotes are also important because it is a line that a function approaches very closely but never reaches. You can find a vertical asymptote where there is division by zero. A horizontal asymptote is “a horizontal line 𝑦 = 𝑏 where the graph approaches the line as the inputs increase or decrease without bound”. Horizontal (and slant) asymptotes therefore correspond to a rational function’s end behavior, which is determined by the ratio of the leading terms of the numerator and denominator. There are three cases that can occur for a horizontal asymptote based on the degrees of the numerator and denominator polynomials. A slant asymptote is defined by a linear equation. If you determine that a rational function has a slant asymptote using the test above, you can find this linear equation using polynomial division. (You can review polynomial division in the week 6 resource.) Divide the rational function’s numerator by its denominator. The quotient (without the remainder) is the equation of the slant asymptote!

Remember to find the domain and the range of functions! The domain is simply what you can put in a function. You should look for red flags like dividing by zero or negative values in square roots. The range is the highest and lowest a function will output. This might be infinity or a simple number like 1.

Highlight #2: Exponents and Logs

Exponential functions model outputs that change at a rate proportional to the current quantity and have the form 𝑓(𝑥) = 𝑎𝑏𝑥

where 𝑎 is the initial value (𝑓 (0)) and 𝑏 is the base or growth factor.

Some characteristics of an exponential function are:

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• Its domain is (−∞, ∞). • Its range is

o (0, ∞) if 𝑎 > 0 o (−∞, 0) if 𝑎 < 0.

• Its 𝑦-intercept is (0, 𝑎). • It has a horizontal asymptote at 𝑦 = 0

When evaluating an exponential function, make sure that you raise 𝑏 to the input value 𝑥 before multiplying by 𝑎 in order to follow the order of operations.

There is one base that is used very frequently in mathematics denoted by the letter 𝑒. This number is defined as follows:

𝑎𝑠𝑛 → ∞, (1 +1𝑛)

" → 𝑒

𝑒 is approximately equal to 2.72. If you ever see lowercase 𝑒 in a function, remember that it is just a constant!

The function 𝑓(𝑥) = 𝑒𝑥 is called the natural exponential.

Graphs of exponential functions with b > 0 look like the following curves:

The inverse of the exponential function 𝑓(𝑥) = 𝑏𝑥 is 𝑓−1 (𝑥) = log𝑏(𝑥). This function is called a logarithm or a logarithmic function, and it has the following definition:

𝑦 = log𝑏(𝑥) is equivalent to 𝑏𝑦 = 𝑥

Because the logarithm is the inverse of the exponential, its domain is the range of the exponential, and its range is the exponential’s domain.

• Domain: (0, ∞) • Range: (−∞, ∞)

There are two logarithms that are frequently used and have special names:

• A common logarithm has a base of 10 and is often written log (𝑥). If you see a logarithm without a base, it is implied that it is a common or base-10 logarithm.

• A natural logarithm has a base of 𝑒 and is written ln (𝑥).

When working with exponentials and logarithms, it is important to know their properties. Your textbook does not have a separate section covering properties of exponential functions, so the following table summarizes both exponential and logarithmic properties. Because exponentials and logarithms are inverses, it is not surprising that their properties also look inverted.

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Another useful tool is the change of base formula which says

log𝑏(𝑀) = log𝑛 (𝑀) log𝑛(𝑏),

where 𝑛 is a new base. The change of base formula can be used to evaluate logarithms in a calculator that only has functions for the common logarithm (log(𝑥)) and the natural logarithm (ln(𝑥)).

Highlight #3: Angles

An angle is “the union of two rays having a common endpoint”. Each ray starts at the endpoint and extends in a straight line from it out to infinity. The endpoint is also known as the angle’s vertex. When we draw angles, it is the convention to draw them in standard position, in which the vertex lies at the origin of the coordinate plane and the initial side lies on the positive x-axis. Also, angles can be positive or negative. Positive angles are measured in the counterclockwise direction, and negative angles are measured in the clockwise direction.

We could prove that one full rotation is 360° or 2𝜋 radians. Therefore,

180° = 𝜋 radians

We can use the above equality to convert between degrees and radians. Note that if an angle measure is not given a unit, it is implied that the units are radians.

Coterminal angles are “two angles in standard position that have the same terminal side”. For a particular angle that is not between 0° (0 radians) and 360° (2𝜋 radians), we often want to find the coterminal angle between 0° (0 radians) and 360° (2𝜋 radians) because this range is easy to work with.

An arc is a portion of the outline of a circle. The formula for arc length 𝑠 is

𝑠 = 𝑟𝜃

where 𝑟 is the radius of the circle that the arc is part of, and 𝜃 is the measure in radians of the angle that forms the arc.

A sector is “a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. See figure 6. To find a sector’s area, we can multiply the whole circle’s area (𝜋𝑟2) by the fraction of the circle that the sector is. This results in the formula:

𝐴 = ½ * 𝜃*r2

Note that 𝜃 must be in radians for the equation to be valid.

Highlight #4: Trig Functions

First of all, it is so important to memorize the unit circle! The unit circle is the circle with radius 1 centered at the origin.

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If we call one of these angles 𝜃, the point (𝑥, 𝑦) at which the terminal ray of the angle intersects the unit circle is given by

𝑥 = cos 𝜃 and 𝑦 = sin 𝜃

𝑓(𝜃) = cos 𝜃 is the cosine function, and 𝑓(𝜃) = sin 𝜃 is the sine function. Note that these functions’ domain is all real numbers, and their range is −1 ≤ 𝜃 ≤ 1.

An important identity relating sine and cosine is the Pythagorean Identity:

cos2𝜃 + sin2𝜃 = 1.

This identity comes from the equation for the unit circle, 𝑥2 + 𝑦2 = 1.

There are four other trigonometric functions: tangent, secant, cosecant, and cotangent. Either a point on the unit circle or the values of sine and cosine for a particular angle can be used to find the values of the other four trigonometric functions.

All trigonometric functions are symmetric in some way. In other words, if we evaluate a trigonometric function at a positive angle and a negative angle of the same magnitude, the absolute values of the two outputs will be the same. Whether each trigonometric function is even or odd can be determined by examining the unit circle. The result of this examination is that cosine and secant are even, while the rest of the trigonometric functions are odd. Therefore, we have the following relationships:

• cos(−𝜃) = cos 𝜃 • sin(−𝜃) = − sin 𝜃 • tan(−𝜃) = − tan 𝜃 • sec(−𝜃) = sec 𝜃 • csc(−𝜃) = − csc 𝜃 • cot(−𝜃) = − cot 𝜃

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Some important applications of trigonometry involve right triangles. So far, we have defined the trigonometric functions in terms of a point on the unit circle, but we can also define them based on the sides of a right triangle. This makes the trigonometric functions much more versatile. The sides of a right triangle are called the hypotenuse, adjacent side, and opposite side. The hypotenuse is always the angle opposite to the right angle. The opposite and adjacent sides, however, vary based on which acute angle in the triangle we want to examine. The opposite side is the side opposite of the acute angle in question. The adjacent side is the side adjacent to (next to) the acute angle in question. See Figure 4.

Knowing these terms for the sides of a right triangle, we can now learn how trigonometric functions of an acute angle 𝜃 are related to these sides.

• sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 • cos 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 • tan 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

A way to remember these relationships is the mnemonic Soh Cah Toa:

• Soh: Sine is opposite over hypotenuse. • Cah: Cosine is adjacent over hypotenuse. • Toa: Tangent is opposite over adjacent.

The sine and cosine functions oscillate smoothly like waves.

A function that can be described as a combination of transformations of the sine or cosine function is called a sinusoidal function or simply a sinusoid. Sinusoids have the general form

𝑦 = 𝐴 sin(𝐵𝑥 − 𝐶) + 𝐷 or 𝑦 = 𝐴 cos(𝐵𝑥 − 𝐶) + 𝐷

Because cosine and sine are shifted versions of each other, the equation for a sinusoid can be written with cosine or sine.

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Because the trigonometric functions are not one-to-one, we have to restrict their domains to a portion where they are one-to-one in order to find their inverses. Our choice of the trigonometric functions’ restricted domains specifies the inverse trigonometric functions’ ranges. By convention, we define the inverse trigonometric functions as shown in the following table.

To determine the value of an inverse trigonometric function, you can use an approach similar to the approach used for logarithmic functions, which are inverses of exponential functions. For example, if you are given an inverse sine function, 𝑦 = sin−1𝑥, ask yourself, “What angle 𝑦 results in sin 𝑦 = 𝑥?” The angle is the output of the inverse function, and it must be in the range shown in Table 1.

We can use the double-angle formulas to rewrite a trigonometric function of a double angle 2𝜃 as a combination of trigonometric functions of the angle 𝜃. We can also use the reduction formulas “to reduce the power of a given expression involving even powers of sine or cosine”. Lastly, we can use the half-angle formulas to rewrite a trigonometric function of a half angle 𝜃/2 with trigonometric functions of 𝜃.

Tips for the Final

1. There is a ton of stuff you learned all semester! I like to split it up in sections, kind of like I did in this document. 2. Make sure to go over old homework. Since there will be a limited amount of time to study, don’t do problems that

you already know how to do. 3. At a certain point, it will be more beneficial to sleep than it will be to study, make sure you take care of

your health for finals week!