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Algebra I - Vocabulary

Vocabulary Word Definition Picture

Area

• The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle.

Axis of Symmetry of a Quadratic

• Divides the parabola into two symmetrical halves • Passes through the vertex • Written as x = #

Axis of Symmetry: x = 5

Binomial • A polynomial of two terms

Ex: 3x + 4 2a2 + 4a

Changes in the “a” of a Quadratic Equation, y = ax2 + c

• Value of “a” determines the width and direction of the parabola Direction: • If “a” is positive, then the parabola opens up • If “a” is negative, then the parabola opens down Width: • As the absolute value of “a” increases, the graph gets narrower • As the absolute value of “a” decreases, the graph gets wider

y = x2 – opens up, y = -x2 – opens down, both have the same width y = 2x2 and y = 5x2 – Both graphs open up but y = 5x2 is narrower than y = 2x2


Changes in the “c” of a Quadratic Equation, y = ax2 + c

• Shifts or translates the parabola up or down • Does NOT change the shape/width of the parabola • When “c” is positive, the graph shifts up • When “c” is negative, the graph shifts down

If the equation y = 2x2 + 3 is shifted up 3 units, the new equation will be y = 2x2 + 6

Changes in the Slope

• Affects the rate of change • Increasing the slope, makes the line steeper • Decreasing the slope makes the line fatter

Changes in the y-intercept

• Changes the starting value, initial value, flat fee, b • Shifts the function up or down

Coefficient

• The number being multiplied to a variable

Ex: -3x2, -3 is the coefficient x2, 1 is the coefficient

Conjecture

• An educated guess based on evidence that is Available

Constant of Variation

• Direct Variation:

• Inverse Variation:


Continuous Function

• A function is continuous when its graph is a single unbroken curve • A graph that you could draw without lifting your pen from the paper

Correlation

• Any relationship between variables in which change in one is related to a change in the other

Decay Factor

• In the equation, , b is the decay factor • b = 1 – r, where r is the rate • 0 < b < 1

Ex: y = 15,000(.85)x b = .85

Dependent Variable

• Is the output value, y, since it depends on the input value it may change

y = 3x + 4, y is the dependent variable C = 4x – 5, C is the dependent variable P = 4v + 7d, P is the dependent variable

Direct Variation

• y = kx, where • A straight line that passes through the origin.

• k is the constant of variation,


Discrete Function

• Discrete Data can only take certain values. • A function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers. • Countable

Ex: The number of students in a class (you can't have half a student)

Distributive Property

• Multiply a number by a sum or difference. (Multiply what is on the outside of the parenthesis to all terms inside the parenthesis.)

Ex: -3(2x + 4) = -3(2x) + -3(4) = -6x – 12

Domain/Input

• The set of all input values, x, or the values of the independent variable, on a graph left to right

f(x) = {(1, 0), (2, 1), (3, 2), (4, 3)} Domain: { 1, 2, 3, 4}

Equation

• A mathematical sentence that states two expression are equal.

Ex: 3x + 4 = 2 4a2 – 4a = 3a + 5

Equation of a Horizontal Line

• y = #, where the slope equals 0

y = 2


Equation of a Vertical Line

• x = #, where the slope is undefined

x = 4

Equation of the Axis of Symmetry

• x = #

Axis of Symmetry: x = 5

Exponent Laws

• When multiplying…. Multiply coefficients, if bases are the same then add exponents • When dividing….Divide coefficients, if bases are the same then subtract the exponents • When raising a power to a power….multiply the exponent on the outside to all of the exponents on the inside • Negative exponents…..Move the base and exponent from the top of the fraction to the bottom (or vice versa) and change the negative exponent to a positive exponent

Ex:

=

= =

=


Exponential Decay

• y = abx, as x increases, y increases

Exponential Growth

• y = abx, as x increases, y decreases

Expression

• A mathematical phrase that can contain numbers, variables, and operation symbols. • The value in an expression can be changed, or “varied”.

Ex: 3x 2x + 4 5a2 + 2a – 1

Function

• A relationship that assigns exactly one output value to each input value • x’s don’t repeat

A function: f(x) = {(1, 0), (2, 1), (3, 2), (4, 3)} Not a function: f(x) = {(1, 0), (1, 1), (3, 2), (4, 3)}, 1’s repeat

Function Notation

• When f(x) is used in place of y to indicate the output values. • Read as “f of x”

can be written as


Graph

• Plots the points of a function on a coordinate plane

Graphing a Linear Function

• Graph two points and connect them with a solid line • Begin with the “b”, and move with the “m”, and then connect the two

Graphing an Inequality

• Graph two points or begin with the “b”, and move with the “m” Connect the Points: • < or > - dashed line

• or ≥ - solid line Shade:

• < or - shade below • > or ≥ - shade above

Growth Factor

• In the equation, , b is the growth factor • b = 1 + r, where r is the rate • b > 1

Ex: y = 15,000(1.05)x b = 1.05


Independent Variable

• Is the input value, x, usually the value you are given

y = 3x + 4, x is the independent variable C = 4x – 5, x is the independent variable P = 4v + 7d, v and d are the independent variables

Inequality

• Composed of two expressions separated by an inequality sign • < - less than • > - greater than

• - less than or equal to, at most, no more than • ≥ - greater than or equal to, at least

Ex: 3x – 4 < 5

Infer

• Make conclusions that must always be true about a function

Intersection Point • A point where lines intersect

Interpret

• Describe a function in terms of its parts

To interpret the relationship, look at how one variable affects the other, For example, does an increase on one lead to an increase or decrease in the other?

Inverse Variation

•

, where

• A curve • As x increases, the y decreases and vice versa • k is the constant of variation,


Like terms

• Terms that have the same variable and the same exponent

Ex: 3x and 4x are like terms 2x2 and -3x2 are like terms 5x and 5 are NOT like terms

Linear Function

• A function whose points lie on a line, greatest power of any variable is 1, can be written in the form f(x) = mx + b, changes at a constant rate, can be represented by equations, tables, graphs, and descriptions • The word “line” is part of “linear”.

Linear Parent Function

• y = x

Graph:

Mapping

• Shows the ordered pairs as two sets of numbers in ovals


Maximum Point

• Point on the graph of a parabola where the vertex is the lowest point

Vertex: (0, 0)

Maximum Point

• Point on the graph of a parabola where the vertex is the highest point

Vertex: (0, 6)

Negative Correlation

• A type of correlation where x increases and y decreases

The more a person brushes, the number of cavities deceases.


Negative Slope

• Falls, decreases, as it goes from the left of a graph to the right

Ordered Pair • (x, y)

Origin

• Where the x and the y-axis intersect • (0, 0)

Parabola

• What the shape of the graph of a quadratic is called • A “u” shaped curve that opens up or opens down

Parallel Lines

• Lines that don’t intersect • Same slope, but different y-intercepts

Ex: The slope, m, is equal to 3 in both equations and their y-intercepts are different; therefore, the lines are parallel.

x

y (0, 0)


Pattern

• A predictable rule that all data in a set follow, often described algebraically

Perimeter • The distance around an object

Perpendicular lines

• Lines that intersect to form a right angle • Slopes are opposite reciprocals

Ex:

3 and

are opposite reciprocal slopes.

Point-Slope Form

• , where m is the slope and is a known point on the line

• Used to write an equation on a line when given a point and a slope • The equation can be rearranged into the slope- intercept form.

Polynomial

• An expression containing one or more terms. • Can be simplified by combining like terms.

Ex: 3x2 – 5x + 6 3x 5


Positive Correlation

• A type of correlation where x increases and y increases

As your height increases, your weight increases

Positive Slope

• Rises, increases, as it goes from the left of a graph to the right

Proportion

• An equation that states two ratios are equal

Quadratic Formula

• a2

ac4bbx

2

• Used to find the solutions to a quadratic equation • Equation must be “= y” or “= 0” to identify the a, b, and c values


Quadratic Function

• A function whose points lie on a parabola, greatest power of any variable is 2, can be

written in the form

Quadratic Parent Function Equation

• y = x2

Range/output

• The set of all output values, y, or the values of the dependent variable, on a graph bottom to top

Range: {-9, -4, -1}

Rate of Change • Aka the slope, “m”, rise/run

Reasonable Solutions

• The values of possible inputs given a function that describes a real-world situation. • Makes common sense in the context of the problem


Roots

• Solutions of a quadratic equation • x-intercepts

Scatter Plot

• A graph that displays the data as ordered pairs. • The pattern of the points shows what relationship, if any, exists between the two variables.

Set Notation

• The ordered pairs of a function are written within brackets

f(x) = {(1, 0), (2, 1), (3, 2), (4, 3)}

Slope

• Steepness of a line • Also called “m” • Also known as the rate of change • Ratio of the vertical change to the horizontal Change

• Can be found using

when given two

points • Can be a positive number, a negative number, 0, or undefined


Slope Formula

•

• used to find the slope when given two points

Ex: Find the slope of a line that contains the points (0, 7) and (5, 3)

Slope-intercept form

• Relates y to x using the slope and y-intercept • Written in the form, y= mx + b • Used to graph an equation when given the “m” and the “b” • Used to write an equation when given the slope and the y-intercept

m = 3 b = -2

Solutions to Inequalities

• Any ordered pair that makes the inequality true • The solutions will be all of the points in the shaded region • Solutions may also be on the line if and only if

the inequality contains a “ or ”

(0, 1) is a solution

(0, -1) is NOT a solution


Solutions to Quadratics

• Where the parabola hits the x-axis • May have 1 solution, 2 solutions, or No solutions • Sometimes called x-intercepts, roots, zeros, or solutions • Calculator can be used – equation must be “= 0” • Enter equation into “y1” • Enter “0” into “y2” • Press “Graph” to see how many solutions • Press “2nd”, “Trace”, #5, “Enter” 3 times • If there are 2 solutions, then repeat “2nd”, “Trace”, move the cursor closer to the other intersection point, and THEN press “Enter” 3 times

Solutions: (-1, 0) and (5, 0)

Solutions to Systems of Equations

• Any ordered pair that is true for all of the equations • 3 types of solutions: Different Lines – 1 Solution, (x, y) Parallel Lines – 0 Solutions, No Solutions Same Lines – Infinitely Many Solutions

Solution: (3, -2) No Solution Infinitely Many

Solving an Inequality

• Very similar to solving an equation • MUST “flip” the inequality sign when multiplying or dividing by a negative (the term being solved for is negative)

Ex:


Solving Quadratics

• When asked to find the solutions of a quadratic, you are looking for the x-intercepts. • Solutions are also called roots, zeros, and x- intercepts.

Solving Systems of Equations

• Find the point that the equations have in common, where the two lines intersect • Calculator can be used to find the solution – both equations must be in slope-intercept form • Enter equations into “y1” and “y2” • Press Graph to see what type of solution • Press “2nd”, “Trace”, “#5” and Enter 3 times

Standard Form • Ax + By = C

System of Equations

• Two or more linear equations using the same variables

Ex: {

Table

• Lists the domain of a function in one column and the range of the function in another column, so that each row contains an ordered pair.


Trend Line/Line of Best Fit

• If there is a correlation between the variables in a scatterplot, the relationship can approximated with a line that shows a trend in the data points.

Trinomial • A polynomial of three terms

Ex: 4x2 – 2x – 4

Undefined Slope

• When the slope, m, is undefined • The line is vertical

Variable

• A symbol, usually a letter, that stands for a number

Vertex

• Minimum point of a parabola if it opens upward • Maximum point of a parabola if it opens downward

Vertex


x-axis

• The horizontal axis on the coordinate plane • Equation of the x-axis: y = 0

x-intercept

• Where a graph intersects the x-axis • Written as (#, 0) • Where the value of the dependent variable, y is equal to 0

y-axis

• The vertical axis on the coordinate plane • Equation of the y-axis: x = 0

y-intercept

• Where a graph intersects the y-axis • Written as (0, #) • Also called “b” • Where the value of the independent variable, x, is equal to 0 • Starting or initial value • Flat fee

Zeros

• Solutions of a quadratic equation • x-intercepts

Zero Slope

• When m = 0 • The line is horizontal

x-intercept: (-2, 0)

y-intercept: (1, 0)

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