Semester 1 Precalculus Review

Unit 1

Distance Formula: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}^{}$ This formula is used to calculate the distance between two points $(x1, y1) and (x2, y2)$

• Midpoint Formula: $M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$ This formula provides the coordinates of the midpoint between points $(x1, y1) and (x2, y2)$ .

• Slope Formula: $m = \frac{y2 - y1}{x2 - x1}$ This formula is essential for determining the slope of the line connecting two points, indicating the line's steepness and direction.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ This formula is used to find the solutions (roots) of a quadratic equation in the form $ax^2 + bx + c = 0$, allowing us to determine the x-intercepts of the corresponding parabola.

How to find x and y-intercepts:

x-intercepts are the points where a graph crosses or touches the x-axis. At these points, the y-coordinate is always zero. To find the x-intercept(s) of an equation, set $y=0$ and solve the equation for $x$.

y-intercepts are the points where a graph crosses or touches the y-axis. At these points, the x-coordinate is always zero. To find the y-intercept(s) of an equation, set x=0 and solve the equation for y

Definitions of symmetry with respect to…

• the x-axis: A graph is symmetric with respect to the x-axis if, for every point $(x,y)$ on the graph, the point $(x,-y)$ is also on the graph. This means the x-axis acts as a mirror, reflecting the upper half of the graph onto the lower half, and vice versa.

  • To test for x-axis symmetry, replace $y$ with $-y$ in the equation. If the resulting equation is equivalent to the original, then the graph is symmetric with respect to the x-axis.

• the y-axis: A graph is symmetric with respect to the y-axis if, for every point $(x,y)$ on the graph, the point $(-x,y)$ is also on the graph. This implies that the y-axis serves as a mirror, reflecting the left half of the graph onto the right half, and vice versa.

  • To test for y-axis symmetry, replace $x$ with $-x$ in the equation. If the resulting equation is equivalent to the original, then the graph is symmetric with respect to the y-axis.

• the origin: A graph is symmetric with respect to the origin if, for every point $(x,y)$ on the graph, the point $(-x,-y)$ is also on the graph. This type of symmetry means the graph remains unchanged after a $180$-degree rotation around the origin.

  • To test for origin symmetry, replace both $x$ with $-x$ and $y$ with $-y$ in the equation simultaneously. If the resulting equation is equivalent to the original, then the graph is symmetric with respect to the origin.

Perfect Square Trinomial:

A perfect square trinomial is a trinomial that results from squaring a binomial. There are two common forms:

• $(a+b)^2 = a^2+2ab+b^2$

• $(a-b)^2 = a^2-2ab+b^2$.

• Example: $(x+3)^2 = x^2+2(x)(3)+3^2 = x^2+6x+9$

Difference of Squares:

The difference of squares is a binomial that can be factored into two binomials, one being the sum of the square roots and the other being the difference of the square roots.

• $(a^2-b^2) = (a+b)(a-b)$

• Example: $(x^2-16) = (x+4)(x-4)$

Lines:

• Equations:

  • Slope Formula: $m = \frac{y2 - y1}{x2 - x1}$

  • Slope-Intercept Form: $y = mx + b$

  • Point-Slope Form: $y - y1 = m(x - x1)$ - This form is ideal when you know the slope $m$ of a line and a single point $(x1, y1)$ that the line passes through

  • General Form: $Ax + By = C$ - In this form, $A$, $B$, and $C$ are real numbers, with $A$ and $B$ not both zero. This form can represent any straight line, including vertical lines, which cannot be represented in slope-intercept form.

• Parallel Lines:

  • Two distinct lines are parallel if and only if they have the exact same slope ($m1 = m2$) and different y-intercepts. This means they will never intersect.

• Perpendicular Lines:

  • Two lines are perpendicular if and only if their slopes are opposite reciprocals of each other. If one line has a slope of $m$, a line perpendicular to it will have a slope of $-\frac{1}{m}$ (provided $m \neq 0$). The product of their slopes is always $-1$. They intersect at a $90$-degree angle.

• Horizontal and Vertical Lines:

  • Horizontal Lines:

    • Have a slope of zero ($m = 0$). This indicates no vertical change for any horizontal change.

    • Their equation is always in the form $y = k$, where $k$ is a constant representing the y-coordinate through which the line passes.

    • Example: $y = 2$

  • Vertical Lines:

    • Have an undefined slope ($x2-x1=0$). This is because there is no horizontal change, making the denominator of the slope formula zero, which is mathematically undefined.

    • Their equation is always in the form $x = k$, where $k$ is a constant representing the x-coordinate through which the line passes.

    • Example: $x = 2$

Circles:

• Definition: A circle is a set of all points in a plane that are equidistant from a fixed point called the center $(h,k)$. This fixed distance is known as the radius $(r)$.

• Standard Form of a Circle: $(x-h)^2+(y-k)^2=r^2$

  • $(h,k)$ represents the coordinates of the center of the circle, and $r$ represents the length of its radius

• Unit Circle: $x^2+y^2=1$

• General Form of a Circle: $x^2+y^2+Ax+By+C=0$

• Going from General Form to Standard Form of a Circle:

  • The process involves completing the square for both the x and y terms.

  • Step 1: Group the x-terms and y-terms together, and move the constant term to the other side of the equation.

  • For example, if you have $x^2+y^2+Ax+By+C=0$, rearrange it to $(x^2+Ax) + (y^2+By) = -C$.

  • Step 2: Complete the square for the x-terms and for the y-terms.

  • To complete the square for $x^2+Ax$, add $(\frac{A}{2})^2$ to both sides of the equation. This turns $(x^2+Ax + (\frac{A}{2})^2)$ into $(x + \frac{A}{2})^2$.

  • Similarly, for $y^2+By$, add $(\frac{B}{2})^2$ to both sides. This turns $(y^2+By + (\frac{B}{2})^2)$ into $(y + \frac{B}{2})^2$.

  • Step 3: Rewrite the expressions as perfect squares and simplify the constant on the right side to find $r^2$.

  • The equation will become $(x + \frac{A}{2})^2 + (y + \frac{B}{2})^2 = -C + (\frac{A}{2})^2 + (\frac{B}{2})^2$. From this, you can identify $h = -\frac{A}{2}$, $k = -\frac{B}{2}$, and $r^2 = -C + (\frac{A}{2})^2 + (\frac{B}{2})^2$.

• Going from Standard Form to General Form of a Circle:

  • This conversion is straightforward and involves expanding the squared binomials.

  • Step 1: Start with the standard form: $(x-h)^2+(y-k)^2=r^2$.

  • Step 2: Expand the binomials:

  • $(x-h)^2 = x^2 - 2hx + h^2$

  • $(y-k)^2 = y^2 - 2ky + k^2$

  • Step 3: Substitute the expanded terms back into the equation and move the $r^2$ term to the left side to set the equation to zero.

  • $x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0$

  • Step 4: Rearrange the terms to match the general form: $x^2+y^2-2hx-2ky+(h^2+k^2-r^2)=0$.

  • Here, $A = -2h$, $B = -2k$, and $C = h^2+k^2-r^2$.

• Area of a circle: $A=\pi r^2$

• Circumference: $C=2\pi r$

  • This formula calculates the distance around the circle (its perimeter), where $r$ is the radius.

• How to find the area of a circumscribed square using the equation of a circle that surround the square???

Unit 2

Functions and their Graphs

• Definition of Functions: let X & Y be two non-empty sets. A function from X into Y is a relation that associates each element of X with exactly one element of Y.

  • If y=f(x), than x is the independent variable (referred to as the [argument of f]) and y is the dependent variable (referred to as the [image of x])

• Function Operations:

  • $(f+g)(x) = f(x) + g(x)$

  • $(f-g)(x) = f(x) - g(x)$

  • (f$\cdot$g)(x) = f(x) $\cdot$ g(x)

  • $(f/g)(x) = f(x)/g(x)$ —> g(x) can’t equal 0

• Difference Quotient:

  • $\frac{\left.f\left(x+h\right)-f\left(x\right.\right)}{h}$

    where h can’t equal zero

  • It is necessary to have a factor of h cancel out of the denominator

  • Step 1 - replace$(x+h)$ for $x$ in your equation for first part, plug in regular equation for $f(x)$

    • $f(x) = 2x²-3x$ —> $\frac{\left(2\left(x+h\right)^2-3\left(x+h\right)\right)-2x^2-3x}{h}$

  • Step 2 - simplify

    • $\frac{h\left(4x+2h-3\right)}{h}$

  • Step 3 - Get rid of the h downstairs

    • $4x+2h-3$ (Difference quotient)

• Finding the Domain of a Function:

  • $f(x) = x²+5x$ —> domain is all reals

  • g(x) = $\frac{3x}{x^2-4}$ (when there’s a variable downstairs it can’t equal zero)

    • $x^2-4=\left(x+2\right)\left(x-2\right)$ —> domain: x can’t equal 2 or -2

  • h(t) = $\sqrt{4-3t}$

    • radicand can’t be negative

    • $4-3t$ must be greater than or equal to 0 —> t has to be less than or equal to 4/3

• Properties of Functions

  • Even Functions:

    • symmetric to the y-axis

    • $f(-x)=f(x)$

  • Odd Functions:

    • symmetric to the origin

    • $f(-x) = -f(x)$

  • Intervals of Change:

    • As x increases…

      • an increasing interval gets larger values of f(x) if x1<x2, then f(x1)<f(x2)

      • a decreasing inte4rval gets smaller values of f(x) if x1<x2, then f(x1)>f(x2)

      • a constant interval retains the same value of f(x) if x1<x2, then f(x1)=f(x2)

  • Local Max/Min:

    • a function f has a local maximum at c if there is an open interval containing c so that $f\left(c\right)$ is greater than or equal to f(x) for all x on the interval and vice versa for minimums

  • Absolute Max/Min:

    • a function F has an absolute maximum at u if f(u) is greater than or equal to f(x) for all x values in the domain of f

  • Extreme Value Theorem:

    • if a function f is continuous on a closed interval [a,b], then f has an absolute maximum and absolute minimum on [a,b]

    • Think of it like this: if you draw a continuous line on a piece of paper without lifting your pen, the highest and lowest points you reached on that line are guaranteed to exist

  • The Library of Functions:

    • $f\left(x\right)=\sqrt{x}$

      

    • no local min/max

  • $f\left(x\right)=^3\sqrt{x}$

    

    • no local min/max or extrema

  • $f\left(x\right)=\left\vert x\right\vert$

    

    • either an absolute max or min

  • f(x)= x

  • 
    

    

    • linear (identity function)

    • no local or absolutes

  • f(x) = x²

    

    • 1 absolute and local

  • $f\left(x\right)=\frac{1}{x}$

    

    • x can never equal 0

    • symmetric to the origin

• Graphing Transformations

  • Vertical shifts

    • $y=f(x)$ —> $y=f(x) + a$

    • Adding to the OUTPUT (doesn’t go with x variable)

    • adding = up, subtracting = down

  • Horizontal shifts

    • $y=f(x)$ —> $y=f(x+a)$

    • adding to the INPUT (goes with x variable)

    • adding = goes to the left, subtracting = goes to the right

  • Vertical stretch

    • $y = f(x)$ —> $y=af(x)$

    • multiplying the OUTPUT (y values)

  • Vertical compression

    • $y = f(x)$ —> $y = af(x)$

      • multiplying the output by $0<a<1$

  • Horizontal stretch & compression

    • $y = f(x)$ —> $y = f(ax)$

    • multiplying the INPUT (x values) by a constant

• Building Functions/Models

  • Given a linear function F and the initial value and rate of change, evaluate $f\left(c\right)$ $ f©

    • 1 - determine the initial value and the slope

    • 2 - substitute the values into f(x) = mx + b

    • 3 - evaluate the function at x = c

  • Expressing the area as a function of x:

    • Area of a rectangle is A = xy

    • A(x) = x times whatever the given equation is

  • If you want to maximize area, substitute in the constraint (e.g. y=50-x)

Unit 3

3.1 —> easy, all about linear functions and their slopes (basic algebra)

• know how to graph linear lines and write linear functions based off word problems

3.2 —> scatterplots (review how to input date into calc)

• The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables, and its value always falls between -1 and +1, inclusive, with 1 being perfect positive, -1 being perfect negative, and 0 meaning no linear correlation. 

3.3 + 3.4

• A quadratic function is a function of the for f(x) = ax²+bx+c$, where a doesn’t equal zero and a, b, c are all real numbers</p></li><li><p>Revenue can be calculated by multiplying the amount of units sold by the price of each unit ($R=xp$)</p></li><li><p>Vertex form of a quadratic:</p><ul><li><p>$f\left(x\right)=a\left(x-h\right)^2+k$</p><p>    (h,k) is the vertex</p></li><li><p>h =$\frac{-b}{2a}$and k =$\frac{\left(4ac-b^2\right)}{4a}$or you can plug in h and get out k</p></li></ul></li><li><p>The axis of symmetry is$x=\frac{-b}{2a}$</p></li><li><p>Discriminant of quadratic equation =$\sqrt{b^2-4ac}$</p><ul><li><p>if d<0 —> no real solutions</p></li><li><p>if d>0 —> two different x-intercepts/solution</p></li><li><p>if d=0 —> only one intercept/solution</p></li></ul></li><li><p>To write the equation of a parabola, you need the vertex and some other point</p><ul><li><p>Step 1 - plug in vertex into vertex form</p></li><li><p>Step 2 - plug in intercept/other point for x and f(x)</p></li><li><p>Step 3 - solve for a</p></li><li><p>Step 4 - plug in a and (h,k) into vertex form and simplify to a quadratic</p></li></ul></li><li><p>A parabola is concave up if <span>a>0</span>; the vertex is the minimum point.</p></li><li><p>A parabola is concave down if <span>a<0</span>; the vertex is the maximum point.</p></li><li><p>To find maximum height of a projectile motion problem —> plug in vertex, output it max height</p></li></ul><p>3.5</p><ul><li><p>Quadratic Inequalities </p><ul><li><p>To solve the inequality$ax^2+bx+c>0 —> graph the quadratic function and from the graph determine where the function is above the x-axis (where f(x)>0)

• To solve:

  • Step 1 - find all intercepts and vertex

  • Step 2 - graph the function

  • Step 3 - identify which way the inequality sign is facing and identify where that is on the graph

  • Step 4 - write the interval for which makes the inequality true]

  • Note: look at concave up or down, if it’s above or below x-axis and sometimes it could result in an empty set if there is no graph in the section you identified

Unit 4

Polynomial and Rational Functions

• 4.1 Polynomial Functions

  • A polynomial function in one variable is a function of the form: f\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$</p><p>where$a_{n},a_{n-1},\ldots,a_1,a_0$are constants, called the coefficients of the polynomial, n is greater than or equal to 0 is an integer, and x is a variable. If$a_{n}$doesn’t equal 0, it is called the leading coefficient, and$n$is the degree of the polynomial</p></li><li><p>The domain of a polynomial function is the set of all real numbers</p></li><li><p>The graphs of polynomials are smooth and continuous</p></li><li><p>The y-intercept is$a_0$</p></li></ul></li><li><p>Power Functions</p><ul><li><p>A power function of degree$n$is a monomial function of the form:$f\left(x\right)=ax^{n}$where a is a real number, doesn’t equal 0 and n>0 is an integer</p></li><li><p>If the power function has an even degree, it’s end behavior is the same as a parabola</p></li><li><p>If the power function has an odd degree, it’s end behavior is the same as$x^3$(as x goes to negative infinity and infinity, y does the same thing)</p></li></ul></li><li><p>Maximum number of turning points (local max/mins) is n - 1</p></li><li><p>Polynomials have non-negative integer exponents associated with their variables —> no variable in the denominator/radicals</p></li><li><p>The following statements are equivalent:</p><ul><li><p>r is a real zero of polynomial f</p></li><li><p>r is an x-intercept of the graph of f</p></li><li><p>x - r is a factor of polynomial f</p></li><li><p>r is a real solution of the equation f(x) = 0</p></li></ul></li><li><p>Polynomial functions with$\left(x-r\right)^{n} as a factor, have r as a real zero of multiplicity n

    • example: f(x) = (x+3)(x+4)³

      • real zeros: -3, -4 with multiplicity of 3

  • If n is even, the graph turns at the intercept (bounces off the x-axis)

  • If n is odd, the graph passes through the intercepts (crosses the x-axis)

  4.2 —> Graphing Polynomial Functions

  • Step 1 - identify the power function —> end behavior

  • Step 2 - identify the intercepts (x & y)

  • Step 3 - identify the real zeros and their multiplicity

  • Step 4 - Identify the possible number of turning points

  • Step 5 - Identify points on the graph both outside and between the x-intercepts

  • Note: If you have a calculator you can approximate the local max/min and the x-intercepts

  4.3 —> Properties of Rational Functions

  • A rational function takes the form of: R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$where p and q are polynomial functions</p></li><li><p>The domain of R is all real numbers except those that make the denominator (q(x)) = 0 (the real zeros of q)</p></li><li><p>Vertical Asymptotes:</p><ul><li><p>The graph may NEVER cross a vertical asymptote</p></li><li><p>The zeros of the denominator of the function fully reduced, and the multiplicities of those zeroes are the vertical asymptotes</p></li></ul></li><li><p>Horizontal Asymptotes</p><ul><li><p>Affect the end behavior of the graph </p></li><li><p>The graph can cross a horizontal asymptote if it is not approaching$\pm\infty$(SOMETIMES)</p></li><li><p>let$R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$where p is degree n, q is degree m</p><ul><li><p>if n<m then y=0 is the asymptote</p></li><li><p>if n=m then y= the ratio of the leading coefficients p/q</p></li><li><p>if n=m+1 then y= the quotient of polynomials (oblique/slant asymptote)</p></li><li><p>if n$\ge$m+2 then <span style="font-family: "Google Sans", Roboto, Arial, sans-serif;"><span>the function's end behavior will resemble the graph of a polynomial —> use polynomial long division (parabola asymptote)</span></span></p></li></ul></li></ul></li><li><p>Polynomial Long Division:</p><ul><li><p>1) divide first term inside by first term outside then put it on top</p></li><li><p>2) multiply top term by both outside terms</p></li><li><p>3) repeat with new terms (divide, multiply, subtract)</p></li><li><p>If remainder put divisor under remainder and add it to answer</p></li></ul></li><li><p>Graphing Rational Functions:</p><ul><li><p>Step 1 - Factor top and bottom and set to lowest terms (if necessary)</p></li><li><p>Step 2 - Find intercepts with multiplicities (DENOMINATOR)</p></li><li><p>Step 3 - Find vertical asymptotes (DENOMINATOR)</p></li><li><p>Step 4 - Find horizontal/slant asymptotes (end behavior)</p><ul><li><p>DEGREES OF NUMERATOR AND DENOMINATOR</p></li></ul></li><li><p>Step 5 - Use x-intercepts to create intervals to find more points</p></li><li><p>Step 6 - Check to see if the graph crosses horizontal asymptote</p><ul><li><p>set the asymptote (y=a) equal to the function and if you get a “valid” x, the graph crosses at (x,a)</p></li></ul></li><li><p>HOLE: when (in fully factored form) there is the same factor on the top and bottom</p></li></ul></li><li><p>Real Zeros of Polynomials</p><ul><li><p>The Remainder Theorem:</p><ul><li><p>suppose f is a polynomial function. if f(x) is divided by (x-c), then the remainder with be$f\left(c\right)$</p></li><li><p>Example: if you divide$x³-4x²-5$by$x-3$, the remainder will be -14, so$f(3)=-14$</p></li></ul></li><li><p>The Factor Theorem:</p><ul><li><p>suppose f is a polynomial function. Then x-c is a factor of f if and only if$f\left(c\right)$= 0</p></li></ul></li></ul></li><li><p><strong>A polynomial function cannot have more real zeros than its degree</strong></p></li><li><p>Descarte’s Rule of Signs:</p><ul><li><p>suppose f is a polynomial function written in standard form</p><ul><li><p>The number of positive real zeros of f either equals the number of sign changes of non-zero coefficients or an even number less than that number (e.g. if there’s 5 sign changes, then there are 5, 3, or 1 positive real zeros)</p></li><li><p>The number of negative real zeros of f either equals the number of sign changes of f(-x)</p><ul><li><p>for f(-x), only factors with an odd exponent will change the sign</p></li></ul></li></ul></li></ul></li><li><p>Rational Zeros Theorem</p><ul><li><p>let f be a polynomial function of degree 1 or higher:</p><ul><li><p>If p/q, in lowest terms, is a rational zero of f, then p is a factor of the constant, a is a factor of leading</p><ul><li><p>essentially its the factors of the constant term over the factors of the leading coefficient ($\frac{p}{q} )