Equations, Functions, and Trigonometry

Equations and Solutions

• The study of everything begins with equations.
• Every equation has a solution set, which includes the values that make the equation true.
• Solving an equation involves finding all the values that satisfy it.
  • For example, the solution to the equation $2x + 1 = 0$ is $x = -\frac{1}{2}$.

Linear Equations

• Have the form $ax + b = 0$, where $a$ and $b$ are constants.
• To solve, isolate $x$ on one side of the equation.
  • Example: Solving $2x + 1 = 0$
    1. Subtract 1 from both sides: $2x = -1$
    2. Divide both sides by 2: $x = -\frac{1}{2}$
• Equations can have no solutions or infinitely many solutions.
  • No solution: $0x = 5$ has no solution because no value of $x$ can make this true.
  • Infinitely many solutions: $0x = 0$ is true for every value of $x$.

Quadratic Equations

• Have the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.
• Usually solved by factoring or using the quadratic formula.
• Factoring involves breaking the quadratic expression into smaller parts that can be solved separately.
  • Example: Solve $x^2 + 5x + 6 = 0$
    • Factor: $(x+2)(x+3) = 0$
    • Solutions: $x = -2$ and $x = -3$
• When factoring is not straightforward, use the quadratic formula:
  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • The formula provides the values of $x$ that satisfy the quadratic equation by plugging in the coefficients a, b, and c.

Equations with Square Roots

• Require special care when solving to avoid extraneous solutions.
• To solve, square both sides to eliminate the radical, but always check the solutions in the original equation.
  • Example: Solve $\sqrt{x+2} = x - 4$
    1. Square both sides: $x + 2 = (x - 4)^2$
    2. Expand and rearrange: $x^2 - 9x + 14 = 0$
    3. Factor: $(x - 7)(x - 2) = 0$
    4. Potential solutions: $x = 7$ and $x = 2$
    5. Check solutions in the original equation:
       • For $x = 7$, $\sqrt{7+2} = 7 - 4$ is true, so $x = 7$ is a valid solution.
       • For $x = 2$, $\sqrt{2+2} = 2 - 4$ is false, so $x = 2$ is an extraneous solution.
• Therefore, the only valid solution is $x = 7$.

Polynomial Equations

• Involve higher powers of $x$, such as $4x^3 + 8x^2 + x + 2 = 0$.
• Can be difficult to solve, but sometimes can be factored using various techniques.
• Solving polynomial equations often involves finding the roots or zeros of the polynomial.

Inequalities

• Similar to equations, but use comparison signs instead of an equal sign (e.g., $\gt, \lt, \geq, \leq$).

Linear Inequalities

• To solve a linear inequality, isolate $x$ while maintaining the inequality.
  • Example: Solve 2x - 3 > 7
    1. Add 3 to both sides: 2x > 10
    2. Divide both sides by 2: x > 5
• Express the solution in interval notation.
  • For x > 5, the interval notation is $(5, \infty)$, indicating all values greater than 5.

Polynomial Inequalities

• Involve polynomials and inequality signs.
• To solve, first factor the polynomial and find the zeros.
  • Example: Solve x^2 - 4 < 0
    1. Factor: (x - 2)(x + 2) < 0
    2. Find zeros: $x = -2$ and $x = 2$
• Create test regions based on the zeros and test points within each region to see where the inequality is true.
  • Test regions: $x < -2$, $-2 < x < 2$, and $x > 2$
  • The inequality is true for -2 < x < 2.

Functions

• A rule that assigns each input to a unique output.
• Written as $f(x) = \text{something}$, where $x$ is the input and $f(x)$ is the output.
• Functions are similar to equations, but with $f(x)$ instead of zero on one side.
  • Example: If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$
• Every function has a domain (valid inputs) and a range (possible outputs).

Graphing Functions

• Plotting $x$ and $f(x)$ on a coordinate grid helps visualize functions.

Linear Functions

• Have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• When graphed, linear functions form a straight line.
  • The slope $m$ indicates how steep the line is.
  • The y-intercept $b$ is the point where the line crosses the y-axis.

Quadratic Functions

• Written as $f(x) = ax^2 + bx + c$.
• Form parabolas, which are U-shaped curves.
• To plot, find the vertex (tip of the parabola) using the formula $x_{\text{vertex}} = -\frac{b}{2a}$, and then find surrounding points by plugging in values into the function.

Circles

• Have the form $(x - a)^2 + (y - b)^2 = r^2$, where $(a, b)$ is the center and $r$ is the radius.

Ellipses

• Stretched circles with the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (centered at the origin).
• $a$ is the length of the ellipse along the x-axis, and $b$ is the length along the y-axis.

Hyperbolas

• Look like two mirrored parabolas.
• Have the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
• To graph, plot the lines $f(x) = \frac{b}{a}x$ and $f(x) = -\frac{b}{a}x$, and then draw the two parabola-looking curves.

Conic Sections

• Quadratics, circles, ellipses, and hyperbolas.
• Called conic sections because they are slices of a cone.

Rational Functions

• The ratio between two polynomials.
  • Example: $f(x) = \frac{3x^2 + 1}{2x^3 + x + 1}$
• Graphs may have vertical asymptotes where the denominator is zero.
• Horizontal or slant asymptotes may also appear based on the polynomial degrees.

Function Properties

• Functions may have symmetry about the y-axis or the origin.

Symmetry About the Y-Axis

• $f(x) = f(-x)$ (e.g., $f(2) = f(-2)$).

Symmetry About the Origin

• $f(-x) = -f(x)$.
• Recognizing symmetry helps efficiently sketch graphs.

Function Transformations

• Replacing $x$ with $x - a$ shifts the graph to the right by $a$ units.
• Replacing $x$ with $x + a$ shifts the graph to the left by $a$ units.
• Adding a constant to the end of a function moves the graph up, while subtracting moves it down.
• Multiplying the function by a number greater than one stretches it vertically, while multiplying by a number between zero and one compresses it vertically.
• Replacing $x$ with a number times $x$ affects the graph horizontally. If the number is greater than one, the graph gets narrower. If it's between zero and one, then the graph gets wider.
• A negative in front of the function reflects it over the x-axis, and a negative inside the function reflects it over the y-axis.

Operations on Functions

• Functions can be added, subtracted, multiplied, and divided.
  • Example: If $f(x) = 3x^2$ and $g(x) = 5x$, then $(f + g)(x) = 3x^2 + 5x$.

Function Composition

• Putting one function inside another.

Inverse Functions

• An inverse function undoes what the original function does.
• If the function takes an input $x$ and gives an output $y$, the inverse takes that $y$ and brings it back to $x$.
• Graphically, the inverse is the reflection across the line $y = x$.
  • Example: If $f(2) = 5$, then $f^{-1}(5) = 2$.

Polynomials

• Functions that can be expressed as a sum of the powers of $x$.
  • Example: $f(x) = 5x^5 + 2x^2 + 1$
• The degree is the highest power in the function, and the coefficient in front of that term is called the leading coefficient.

Trigonometry

The Unit Circle

• A circle with a radius of one.
• Imagine a point on that circle starting on the positive x-axis and rotating around in a counterclockwise direction.
• Use $\theta$ to denote the angle.
• Convention: start from the positive x-axis and go counterclockwise.
• Create a right triangle by considering the vertical and horizontal components of my particular point.

Trig Functions

• Define the horizontal and vertical components to be cosine of $\theta$ and sine of $\theta$, respectively.
• In a right triangle, sine is the opposite over hypotenuse, and cosine is the adjacent over the hypotenuse.
• For a unit circle, the hypotenuse is always one, so adjacent and opposite are cosine and sine, respectively.
• As the angle $\theta$ increases, the sides of the triangle can point in either the positive or negative directions with respect to the x and y axes.
• Sine of $\theta$ and cos of $\theta$ are sometimes positive and sometimes negative.
• In calculus, we pretty much always use radians for the angle of $\theta$.

Radians

• If I imagine going the entire way around the circle, this is called $2\pi$ radians.
• The circumference of a circle is $2\pi$ times the radius.
• If it's a unit circle, then it's just $2\pi$ times one or $2\pi$.
• So the the reason I like radians so much is that my circumference is $2\pi$ and the angle is $2\pi$, they exactly match.
• Or or if I just take a portion of it, like like, for example, suppose I do a quarter of a circle, this is $\theta$ going to $2\pi$ divided by four or $\frac{\pi}{2}$, and then the arc length from that quarter circle, also $\frac{\pi}{2}$.
• There's this correspondence when you use radians.
• The consequence of this in calculus is that it will avoid a bunch of weird stretching factors appearing.
• If you try to do things in degrees, you're gonna have all these $2\pi$ divided by 360 degree factors appearing. It's really nice when you use radians.

Pythagorean Trigonometric Identity

• A normal right triangle, with sides a, b, and c, has the Pythagorean theorem that $a^2 + b^2 = c^2$.
• In our specific context where we have cos and sine, I get that $cos^2 + sin^2$ is the hypotenuse squared or one.
• $cos^2(\theta) + sin^2(\theta) = 1$

Graphs of Sine and Cosine

• Cosine of $\theta$ is thought of as the horizontal component.
• If I just record that horizontal component as I rotate along, then I just get the graph of cosine of $\theta$.
• When $\theta$ is zero, cosine of $\theta$ is going to be one.
• But then as I rotate around to $\frac{\pi}{2}$, that horizontal component shrinks down to zero.
• Rotating further to $\pi$, it starts increasing now in the negative direction.
• So the graph goes down to negative one.
• Rotating to $\frac{3\pi}{2}$, it's going to go back to zero and then finally stretches back out to one as you get around to $2\pi$.
• Sine of $\theta$ is the vertical component as the point rotates around the circle.
• This will give the graph of sine.
• You can always generate the sine and cosine graphs and answer questions like cosine of $\frac{3\pi}{2}$ just by knowing the unit circle.
• The two graphs look really similar to each other, just sort of shifted over -- captured by another identity such as cosine of $\theta$ is just sine of $(\theta + \frac{\pi}{2})$.

Special Triangles

• The hypotenuse has length one, but I don't know what the other side is gonna be; let's just call it something generic like a.
• $\frac{\pi}{4}$ is cutting things exactly in half; I have to have this isosceles triangle here by Pythagoras.
• $a^2 + a^2 = 1$.
• So $2a^2 = 1$, then I solve, this gives the value of a equal to $\frac{1}{\sqrt{2}}$.
• If I multiply this triangle by a factor of $\sqrt{2}$, it gives me 1 1 $\sqrt{2}$.

Solving for Special Values

• Sine of $\frac{\pi}{4}$ = $\frac{1}{\sqrt{2}}$.

Other Trig Functions

• Basic trig Functions:
  • Tangent
  • Cotangent
  • Secant
  • Cosecant.
• Six ways you can take a ratio of one of those to one of the other ones, and each of those six ways is given a name.

Relation to Original Sine and Cosine

• Tangent: $\frac{sine}{cosine}$
• Cotangent: $\frac{cosine}{sine}$
• Secant : $\frac{1}{cosine}$
• Cosecant: $\frac{1}{sine}$
• In calculus, this is really, really useful as we start taking derivatives of one of the other functions.

Graph of Tangent

• Whenever cosine is zero, which you'll notice happens at $\frac{\pi}{2}$, $\frac{3\pi}{2}$, $-\frac{\pi}{2}$, and so forth, any places where that denominator is zero, it has to have a vertical asymptote.
• In the numerator, which is sine, sine which is the blue graph is zero at zero and $\pi$ and $2\pi$ and negative $\pi$ and so forth.

Tangent Graph Analysis

• The vertical asymptote is $\pi$ over two, and the zeros are gonna be the matter of connecting, like, they go up to positive infinity and they go down to negative infinity and say in zero to $\frac{\pi}{2}$, we know that it has to go to positive infinity because sine divided by cos in this region, the graph of cosine is positive, the graph of sine is positive, so the tangent has to be positive as well.
• Whereas, for example, between $\frac{\pi}{2}$ and $-\frac{\pi}{2}$, my sine is positive.
• My cosine is negative, so sine divided by cos better be negative, and that's why tangent starts at negative infinity and goes up to zero.

Pythagorean Trigonometric Identity analogs

• Dividing $cos^2(\theta) + sin^2(\theta) = 1$ by sine squared:
  • Gives: $1 + cot^2(\theta) = csc^2(\theta)$.
• or if instead I prefer to divide by cosine squared, I get tan squared plus one equals secant squared.
• $tan^2(\theta) + 1 = sec^2(\theta)$.
• Three different pairings, sine and cosine, cotangent and cosecant, tangent and secant, and they really often work together.
• Take whatever complicated expressions I have that are combining lots of things and try to make them all be sines and cosas or all cotangent and cosecants or all tangent and secants. These sort of three pairs will keep on working together.