Math 171 College Algebra Ch. 1 + 2 Midterm #1

midpoint formula

((x₁+x₂)/2, (y₁+y₂)/2) Take the average of each of the coordinates

a²+2ab+b²

(a+b)²

a²-b²

(a-b)(a+b)

a²-2ab+b²

(a-b)²

Standard form of a circle's equation

(x-h)²+(y-k)²=r² x - x-coordinate of the point h - x-coordinate of the center y - y-coordinate of the point k - y-coordinate of the center r - radius

|u| < c

-c<u<c u is greater than -c AND less than positive c

b² - 4ac = 0

1 real number solution. There is only 1 because it repeats, it is the same answer for both scenarios

b² - 4ac < 0

2 imaginary solutions

piecewise function

2 or more equations over a specific domain

b² - 4ac > 0

2 real number solutions

Constant function

A function like f(x) = c that is the same regardless of the independent variable graph is horizontal line. domain - (-∞,∞) range - the single value c constant on (-∞,∞)

one to one function

A function where each element of the range is paired with exactly one element of the domain. Ex. f(x)=x+1 different inputs will never receive the same answer. Are continually increasing or decreasing but not both Pass a horizontal line test with only one spot

ac method

Method for factoring trinomials by replacing the middle term with two terms that allow us to factor the resulting four-term polynomial by grouping. 1. Find the 2 numbers that multiply to ac AND add to b 2. Replace b with those 2 numbers. Don't forget that they should still have the variable that was attached to b 3. Factor by grouping

stretch/shrink a graph horizontally

divide each of the graph's x coordinates by c I don't know what the reason for doing this is but you need to know it for at least pre-calc so I guess you should learn it

identity function

f(x) = x Domain: (-∞ ,∞) Range: (-∞ ,∞) Increasing on (-∞ ,∞) Odd Function

Standard Quadratic equation

f(x) = x² Domain (-∞,∞) Range [0,∞) Decreasing on (-∞,0) Increasing on (0,∞) Even Function

Cubic Function

f(x) = x³ Domain (-∞,∞) Range (-∞,∞) Increasing on (-∞,∞) Odd Function

Absolute Value function

f(x) = |x| Domain (-∞,∞) Ranges [0.∞) Decreasing on (-∞,0) Increasing on (0,∞) Even function

Cube Root Function

f(x) = ³√x Domain (-∞,∞) Range (-∞,∞) Increasing on (-∞,∞) Odd Function

Square Root Function

f(x) = √(x) Domain [0, ∞) Range [0, ∞) Increasing on (0, ∞) Neither even or odd

find a composition of functions when given their graphs and a number to plug

find the number to plug on each of the graphs and take the y value (f(x)) then add/subtract/multiply/divide (whatever you're told to do in the instructions) the 2 y values. That is your composition of the 2 functions.

odd function

graph is symmetrical with respect to the origin; f(-x) = -f(x) Every term on the right side of the equation Changes signs if x is replaced with -x Symmetric with respect to the origin Every term contains x to an odd power (a constant can be thought of as having an x⁰ which is even. x can be thought of as x¹ which is odd)

even function

graph is symmetrical with respect to the y-axis; f(x) = f(-x) The left side of an equation does not change if x is replaced with -x Y-axis symmetry Every term contains x to an even power (a constant can be thought of as having an x⁰ which is even. x can be thought of as x¹ which is odd)

evaluate composite functions via their graphs

in (a○b)(c) find b(c) from the graph first. Then plug that into the function a wherever c is present. That is the answer

When asked for when the graph is increasing/decreasing

include all the x values while y goes up or down, NOT the y values. Remember: the values are NOT included ("[" or "]") because once you reach that point they are no longer increasing or decreasing.

perpendicular lines

inverse slopes, negative reciprocols of each other, the product of both slopes is -1

remember that 1

is a square so you can factor it.

Find the domain of a function

largest set of real numbers for which the value f(x) is a real number. Exclude any numbers that would cause division by 0 or any numbers that result in a negative under an even root. Ex. f(x) = x²+5x+7 → (-∞,∞) (no denominators or roots) Ex. f(x) = 5x/x²-49 → x²-49=0 → (x-7)(x+7)=0 → x=-7,7 → (-∞,-7)∪(-7,7)∪(7,∞) (1 denominator, no roots) Ex. f(x) = √9x-27 → 9x-27≥0 → 9(x-3)≥0→ x-3≥0/9 → x≥3 → [3,∞) (no denominators, 1 root) Ex. 5x/(√24-3x) → 24-3x>0 → 3(8-x)>0 → 8-x>0/3 → 8>x → (-∞,8) (1 denominator, 1 root)

Solve absolute value inequalities

make 2 scenarios. 1 with the absolute value bars dropped, and the inequality sign stays the same, and the constant stays the same. And 1 with the absolute value bars dropped, the inequality symbol flipped, and the negative of the constant. Ex. |6x-3|<15 2 scenarios 6x-3<15 and 6x-3>-15

stretch/shrink a graph vertically

multiply each of the graph's y coordinates by c I don't know what the reason for doing this is but you need to know it for at least pre-calc so I guess you should learn it

Test for symmetry relative to the origin

replace x and y with -x and -y. If the equation doesn't change it has symmetry relative to the origin. If the equation is an odd function it has symmetry relative to the origin Whenever x² = y²... Or y² = x²... Always check for all types of symmetry! You can have more than 1 or none

parallel lines

same slope

find 0s of f

set f(x) to 0 and find the x value(s) where y is 0

Domain of a function that is the combination of other functions

start with the domain of 1 function and exclude any numbers excluded in the other function's domain.

How to tell if the inverse of a function will be a function from a graph

take the function (it passes a vert line test) and try a horizontal line test on it. If it passes that then its inverse will be a function, if it fails then its inverse is not a function.

How to word a rate of change

the *whatever the y value represents* *increases/decreases* at a rate of *slope* per *whatever x value represents*

Secant line

the line through two points on a curve

Complete the square

the process of adding a term to a quadratic expression to make it a perfect square trinomial 1. Start with an equation simplified to ax² + bx = c. Variable terms on one side, constants on the other. 2. make a equal 1 3. complete the square by adding (b/2)² to both sides. 4. Factor the left side as (x + b/2)² 5. Solve for x with the square root property

|u| > c

u<-c or u>c u is less than -c OR greater than positive c

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

General form of a circle's equation

x² + y² + Dx + Ey + F = 0

reflection in x axis

y = -f(x)

reflection in y axis

y = f(-x)

horizontal shift

y = f(x + c) is the graph of y = f(x) shifted c units left y = f(x - c) is the graph of y = f(x) shifted c units right

vertical shifts

y = f(x) + c is the graph of y = f(x) shifted c units up y = f(x) - c is the graph of y = f(x) shifted c units down

Point-slope form

y-y₁ = m(x-x₁) When told to answer in point-slope form but also to simplify: only simplify - signs. You basically just have to plug in a point and the slope and not distribute or else it's no longer in point-slope form

Find the values for which f(x) ≤ 0 on a graph

​(​minimum x value, maximum x value​]∪​{single x value outside of the interval​}

Convert general form of a circle to standard form

Complete the square 1. Make the coefficients of x² and y² 1 2. rearrange the terms to move xs by xs and ys by ys. And move the constants to the other side 3. Add (b/2)² to both sides for both x and y. b will take on 2 values since there are 2 variables to the 1st degree 4. Factor into (x + (b/2))² + (y + (b/2))² = -F

y is a function of x

Solve for y in terms of x. If there is only 1 possible answer for y it is a function of x

Graph the inverse of a one to one function

Take the points of the original function and switch the x and y coords. Then graph. Ex. there's a function, f, with the points (1,2), (-2,3), and (-5,-6). Graph the inverse by plotting (2,1), (3,-2), and (-6,-5)

radius

The distance from the center of a circle to any point on the circle

relative maxima

The highest point in a particular section of a graph. When asked for the relative maxima give y coord or function value of the point When asked to describe where the FUNCTION has a relative maximum give the x coord

relative minima

The lowest point in a particular section of a graph. When asked for the relative manimum give y coord or function value of the point When asked to describe where the FUNCTION has a relative minimum give the x coord

Domain

The set of x-coordinates in a relation. ex {2,4,6} Duplicate values do NOT get listed multiple times

Range

The set of y-coordinates in a relation ex. {4,2,1} Duplicate values do NOT get listed multiple times

Inverse Function

Two functions f and g are inverse functions if and only if both of their compositions are the identity function To create the inverse of a function convert to the manipulative form of the function (f(x) to y) and switch x and y. Then convert back to function form. An inverse function is notated by the function with a superscript -1: f⁻¹(x) An inverse function undoes the original function. f(g(x)) = x AND g(f(x)) = x THEN Inverse Functions Have inverse graphs (x becomes y while y becomes x for each point) They are reflections of each other anchored on the graph of y=x. So they aren't reflections about the x-axis or the y-axis but have a subtler reflection. The domain becomes the range of the inverse function and the range becomes the domain of the inverse function.

Check if you have the correct amount of answers

Using the discriminate

Function

When 1 number in a domain corresponds to 1 number in the range that relation IS a function. If there are duplicate x values then that is NOT a function but if there are duplicate y values then that can still be a function. x²=y Function y²=x Not a function

Find slope

When given 2 points: (y₁-y₂)/(x₁-x₂) When given equation in slope int form: y=mx+b m is slope (including negative signs) When given equation in point-slope form: y-y₁=m(x-x₁) m is slope (including negative signs) An undefined slope is a vertical line

Test for symmetry relative to the y axis

With an equation replace x with -x and if the equation doesn't change the equation has y-axis symmetry If the equation is an even function it has y-axis symmetry Whenever y¹ = x²... Always check for all types of symmetry! You can have more than 1 or none

Test for symmetry relative to the x axis

With an equation replace y with -y and if the equation doesn't change the equation has x-axis symmetry With a graph for every point (x,y) there is also a point (-x,y) Whenever x¹ = y²... Always check for all types of symmetry! You can have more than 1 or none

Difference Quotient

[f(x+h)-f(x)]/h where h does not = 0 just plug (x+h) into f(x) and then f(x) into f(x). Simplify and that's your answer

Composition of functions

a function is performed, and then a second function is performed on the result of the first function. (f○g)(x) = f(g(x)). f(g(x)) means that function g is the input to function f. Wherever x is in f(g(x)) replace it with (g(x)). and then wherever (g(x)) is in f(g(x)) replace it with g(x) Domain: the set of all x such that a) x is in the domain of g and b) g(x) is in the domain of f Ex. f(x) = 5x+6; g(x) = 2x²-x-1; find (f○g)(x) 5(g(x))+6 → 5(2x²-x-1)+6 → 10x²-5x-5+6 → 10x²-5x+1 D𝒻₍ₓ₎ = (-∞,∞) because there are no denominators or roots D𝓰₍ₓ₎ = (-∞,∞) because there are no denominators or roots. So the domain is (-∞,∞) Ex. f(x)= 5x+6; g(x) 2x²-x-1; find (f○g)(-1) 5(g(-1))+6 → 5(2(-1)²-(-1)-1)+6 → 5(2+1-1)+6 → 5(2)+6 → 10+6 → 16. The domain is (-∞,∞)

Inverse of a function isn't always

a function. Ex. What is the inverse of f(x)=x²-2 Swap x and y. x=y²-2 Solve for y y=±√x+2 Since a function can only have 1 y value for each x value this is not a function, (though, it is the inverse equation of the function)

general form

ax + by + c = 0 a is positive no fractions or decimals

Relation

a set of ordered pairs ex {(2,4),(4,2),(6,1)}

Step Function

a special kind of piecewise function whose graph is a series of line segments

(a+b)²

a²+2ab+b²

(a-b)²

a²-2ab+b²

(ab)²

a²b²

distance between 2 points formula

d = √(x₂-x₁)²+(y₂-y₁)²

To determine if there are any shifts/translations, reflections, stretching, and/or shrinking

Basically subtract the original equation from the new equation and all that's left over is what you have to account for

When given 2 functions and asked to determine if they are inverses or not

Find the composition of the functions and if you are left with just x in both of them they are inverse functions (they have effectively canceled each other out) f(g(x)) = x AND g(f(x)) = x THEN Inverse Functions

To find the average rate of change on a function

Find the slope between the 2 points you are supposed to find the average rate of change for. The only difference from a curve and a line is that this is only the AVERAGE rate of change as opposed to the absolute rate of change and is only for that specific area between the points.

vertical line test

If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.